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Theorem infmap2 9321
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9679 avoids the axiom of choice, it requires the powerset of an infinite set to be well-orderable and so is usually not applicable. (Contributed by NM, 1-Oct-2004.) (Revised by Mario Carneiro, 30-Apr-2015.)
Assertion
Ref Expression
infmap2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem infmap2
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 oveq2 6878 . . 3 (𝐵 = ∅ → (𝐴𝑚 𝐵) = (𝐴𝑚 ∅))
2 breq2 4848 . . . . 5 (𝐵 = ∅ → (𝑥𝐵𝑥 ≈ ∅))
32anbi2d 616 . . . 4 (𝐵 = ∅ → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ≈ ∅)))
43abbidv 2925 . . 3 (𝐵 = ∅ → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
51, 4breq12d 4857 . 2 (𝐵 = ∅ → ((𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ (𝐴𝑚 ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}))
6 simpl2 1237 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵𝐴)
7 reldom 8194 . . . . . . . . . . 11 Rel ≼
87brrelexi 5358 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
96, 8syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
107brrelex2i 5359 . . . . . . . . . 10 (𝐵𝐴𝐴 ∈ V)
116, 10syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12 xpcomeng 8287 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
139, 11, 12syl2anc 575 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14 simpl3 1239 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ∈ dom card)
15 simpr 473 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16 mapdom3 8367 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))
1711, 9, 15, 16syl3anc 1483 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴𝑚 𝐵))
18 numdom 9140 . . . . . . . . . 10 (((𝐴𝑚 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴𝑚 𝐵)) → 𝐴 ∈ dom card)
1914, 17, 18syl2anc 575 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20 simpl1 1235 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21 infxpabs 9315 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
2219, 20, 15, 6, 21syl22anc 858 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23 entr 8240 . . . . . . . 8 (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
2413, 22, 23syl2anc 575 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25 ssenen 8369 . . . . . . 7 ((𝐵 × 𝐴) ≈ 𝐴 → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
2624, 25syl 17 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
27 relen 8193 . . . . . . 7 Rel ≈
2827brrelexi 5358 . . . . . 6 ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∈ V)
2926, 28syl 17 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∈ V)
30 abid2 2929 . . . . . 6 {𝑥𝑥 ∈ (𝐴𝑚 𝐵)} = (𝐴𝑚 𝐵)
31 elmapi 8110 . . . . . . . 8 (𝑥 ∈ (𝐴𝑚 𝐵) → 𝑥:𝐵𝐴)
32 fssxp 6271 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥 ⊆ (𝐵 × 𝐴))
33 ffun 6255 . . . . . . . . . . 11 (𝑥:𝐵𝐴 → Fun 𝑥)
34 vex 3394 . . . . . . . . . . . 12 𝑥 ∈ V
3534fundmen 8262 . . . . . . . . . . 11 (Fun 𝑥 → dom 𝑥𝑥)
36 ensym 8237 . . . . . . . . . . 11 (dom 𝑥𝑥𝑥 ≈ dom 𝑥)
3733, 35, 363syl 18 . . . . . . . . . 10 (𝑥:𝐵𝐴𝑥 ≈ dom 𝑥)
38 fdm 6260 . . . . . . . . . 10 (𝑥:𝐵𝐴 → dom 𝑥 = 𝐵)
3937, 38breqtrd 4870 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥𝐵)
4032, 39jca 503 . . . . . . . 8 (𝑥:𝐵𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4131, 40syl 17 . . . . . . 7 (𝑥 ∈ (𝐴𝑚 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4241ss2abi 3871 . . . . . 6 {𝑥𝑥 ∈ (𝐴𝑚 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
4330, 42eqsstr3i 3833 . . . . 5 (𝐴𝑚 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
44 ssdomg 8234 . . . . 5 ({𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∈ V → ((𝐴𝑚 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}))
4529, 43, 44mpisyl 21 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)})
46 domentr 8247 . . . 4 (((𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)}) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4745, 26, 46syl2anc 575 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
48 ovex 6902 . . . . . . 7 (𝐴𝑚 𝐵) ∈ V
4948mptex 6707 . . . . . 6 (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V
5049rnex 7326 . . . . 5 ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V
51 ensym 8237 . . . . . . . . . . . 12 (𝑥𝐵𝐵𝑥)
5251ad2antll 711 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → 𝐵𝑥)
53 bren 8197 . . . . . . . . . . 11 (𝐵𝑥 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑥)
5452, 53sylib 209 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 𝑓:𝐵1-1-onto𝑥)
55 f1of 6349 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵𝑥)
5655adantl 469 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝑥)
57 simplrl 786 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥𝐴)
5856, 57fssd 6266 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝐴)
5911, 9elmapd 8102 . . . . . . . . . . . . . . 15 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
6059ad2antrr 708 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴𝑚 𝐵) ↔ 𝑓:𝐵𝐴))
6158, 60mpbird 248 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓 ∈ (𝐴𝑚 𝐵))
62 f1ofo 6356 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵onto𝑥)
63 forn 6330 . . . . . . . . . . . . . . . 16 (𝑓:𝐵onto𝑥 → ran 𝑓 = 𝑥)
6462, 63syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝐵1-1-onto𝑥 → ran 𝑓 = 𝑥)
6564adantl 469 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → ran 𝑓 = 𝑥)
6665eqcomd 2812 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥 = ran 𝑓)
6761, 66jca 503 . . . . . . . . . . . 12 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
6867ex 399 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (𝑓:𝐵1-1-onto𝑥 → (𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓)))
6968eximdv 2008 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (∃𝑓 𝑓:𝐵1-1-onto𝑥 → ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓)))
7054, 69mpd 15 . . . . . . . . 9 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
71 df-rex 3102 . . . . . . . . 9 (∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴𝑚 𝐵) ∧ 𝑥 = ran 𝑓))
7270, 71sylibr 225 . . . . . . . 8 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓)
7372ex 399 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥𝐴𝑥𝐵) → ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓))
7473ss2abdv 3872 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓})
75 eqid 2806 . . . . . . 7 (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓)
7675rnmpt 5572 . . . . . 6 ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴𝑚 𝐵)𝑥 = ran 𝑓}
7774, 76syl6sseqr 3849 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
78 ssdomg 8234 . . . . 5 (ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∈ V → ({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓)))
7950, 77, 78mpsyl 68 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
80 vex 3394 . . . . . . . . 9 𝑓 ∈ V
8180rnex 7326 . . . . . . . 8 ran 𝑓 ∈ V
8281rgenw 3112 . . . . . . 7 𝑓 ∈ (𝐴𝑚 𝐵)ran 𝑓 ∈ V
8375fnmpt 6227 . . . . . . 7 (∀𝑓 ∈ (𝐴𝑚 𝐵)ran 𝑓 ∈ V → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴𝑚 𝐵))
8482, 83mp1i 13 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴𝑚 𝐵))
85 dffn4 6333 . . . . . 6 ((𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) Fn (𝐴𝑚 𝐵) ↔ (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
8684, 85sylib 209 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓))
87 fodomnum 9159 . . . . 5 ((𝐴𝑚 𝐵) ∈ dom card → ((𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓):(𝐴𝑚 𝐵)–onto→ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) → ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵)))
8814, 86, 87sylc 65 . . . 4 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵))
89 domtr 8241 . . . 4 (({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴𝑚 𝐵) ↦ ran 𝑓) ≼ (𝐴𝑚 𝐵)) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵))
9079, 88, 89syl2anc 575 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵))
91 sbth 8315 . . 3 (((𝐴𝑚 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ∧ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴𝑚 𝐵)) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
9247, 90, 91syl2anc 575 . 2 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
937brrelex2i 5359 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94933ad2ant1 1156 . . . 4 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → 𝐴 ∈ V)
95 map0e 8126 . . . 4 (𝐴 ∈ V → (𝐴𝑚 ∅) = 1𝑜)
9694, 95syl 17 . . 3 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 ∅) = 1𝑜)
97 1onn 7952 . . . . . 6 1𝑜 ∈ ω
9897elexi 3407 . . . . 5 1𝑜 ∈ V
9998enref 8221 . . . 4 1𝑜 ≈ 1𝑜
100 df-sn 4371 . . . . 5 {∅} = {𝑥𝑥 = ∅}
101 df1o2 7805 . . . . 5 1𝑜 = {∅}
102 en0 8251 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
103102anbi2i 611 . . . . . . 7 ((𝑥𝐴𝑥 ≈ ∅) ↔ (𝑥𝐴𝑥 = ∅))
104 0ss 4170 . . . . . . . . 9 ∅ ⊆ 𝐴
105 sseq1 3823 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
106104, 105mpbiri 249 . . . . . . . 8 (𝑥 = ∅ → 𝑥𝐴)
107106pm4.71ri 552 . . . . . . 7 (𝑥 = ∅ ↔ (𝑥𝐴𝑥 = ∅))
108103, 107bitr4i 269 . . . . . 6 ((𝑥𝐴𝑥 ≈ ∅) ↔ 𝑥 = ∅)
109108abbii 2923 . . . . 5 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = {𝑥𝑥 = ∅}
110100, 101, 1093eqtr4ri 2839 . . . 4 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = 1𝑜
11199, 110breqtrri 4871 . . 3 1𝑜 ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}
11296, 111syl6eqbr 4883 . 2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
1135, 92, 112pm2.61ne 3063 1 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴𝑚 𝐵) ∈ dom card) → (𝐴𝑚 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  Vcvv 3391  wss 3769  c0 4116  {csn 4370   class class class wbr 4844  cmpt 4923   × cxp 5309  dom cdm 5311  ran crn 5312  Fun wfun 6091   Fn wfn 6092  wf 6093  ontowfo 6095  1-1-ontowf1o 6096  (class class class)co 6870  ωcom 7291  1𝑜c1o 7785  𝑚 cmap 8088  cen 8185  cdom 8186  cardccrd 9040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7175  ax-inf2 8781
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-reu 3103  df-rmo 3104  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-int 4670  df-iun 4714  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-se 5271  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6060  df-fun 6099  df-fn 6100  df-f 6101  df-f1 6102  df-fo 6103  df-f1o 6104  df-fv 6105  df-isom 6106  df-riota 6831  df-ov 6873  df-oprab 6874  df-mpt2 6875  df-om 7292  df-1st 7394  df-2nd 7395  df-wrecs 7638  df-recs 7700  df-rdg 7738  df-1o 7792  df-oadd 7796  df-er 7975  df-map 8090  df-en 8189  df-dom 8190  df-sdom 8191  df-fin 8192  df-oi 8650  df-card 9044  df-acn 9047
This theorem is referenced by:  infmap  9679
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