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Theorem infmap2 9636
 Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 9994 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 7148 . . 3 (𝐵 = ∅ → (𝐴m 𝐵) = (𝐴m ∅))
2 breq2 5035 . . . . 5 (𝐵 = ∅ → (𝑥𝐵𝑥 ≈ ∅))
32anbi2d 631 . . . 4 (𝐵 = ∅ → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ≈ ∅)))
43abbidv 2862 . . 3 (𝐵 = ∅ → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
51, 4breq12d 5044 . 2 (𝐵 = ∅ → ((𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ (𝐴m ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}))
6 simpl2 1189 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵𝐴)
7 reldom 8505 . . . . . . . . . . 11 Rel ≼
87brrelex1i 5573 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
96, 8syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
107brrelex2i 5574 . . . . . . . . . 10 (𝐵𝐴𝐴 ∈ V)
116, 10syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12 xpcomeng 8599 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
139, 11, 12syl2anc 587 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14 simpl3 1190 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ∈ dom card)
15 simpr 488 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16 mapdom3 8680 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
1711, 9, 15, 16syl3anc 1368 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
18 numdom 9456 . . . . . . . . . 10 (((𝐴m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴m 𝐵)) → 𝐴 ∈ dom card)
1914, 17, 18syl2anc 587 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20 simpl1 1188 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21 infxpabs 9630 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
2219, 20, 15, 6, 21syl22anc 837 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23 entr 8551 . . . . . . . 8 (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
2413, 22, 23syl2anc 587 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25 ssenen 8682 . . . . . . 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 8504 . . . . . . 7 Rel ≈
2827brrelex1i 5573 . . . . . 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 2932 . . . . . 6 {𝑥𝑥 ∈ (𝐴m 𝐵)} = (𝐴m 𝐵)
31 elmapi 8418 . . . . . . . 8 (𝑥 ∈ (𝐴m 𝐵) → 𝑥:𝐵𝐴)
32 fssxp 6511 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥 ⊆ (𝐵 × 𝐴))
33 ffun 6493 . . . . . . . . . . 11 (𝑥:𝐵𝐴 → Fun 𝑥)
34 vex 3444 . . . . . . . . . . . 12 𝑥 ∈ V
3534fundmen 8573 . . . . . . . . . . 11 (Fun 𝑥 → dom 𝑥𝑥)
36 ensym 8548 . . . . . . . . . . 11 (dom 𝑥𝑥𝑥 ≈ dom 𝑥)
3733, 35, 363syl 18 . . . . . . . . . 10 (𝑥:𝐵𝐴𝑥 ≈ dom 𝑥)
38 fdm 6498 . . . . . . . . . 10 (𝑥:𝐵𝐴 → dom 𝑥 = 𝐵)
3937, 38breqtrd 5057 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥𝐵)
4032, 39jca 515 . . . . . . . 8 (𝑥:𝐵𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4131, 40syl 17 . . . . . . 7 (𝑥 ∈ (𝐴m 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4241ss2abi 3994 . . . . . 6 {𝑥𝑥 ∈ (𝐴m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
4330, 42eqsstrri 3950 . . . . 5 (𝐴m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
44 ssdomg 8545 . . . . 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 8558 . . . 4 (((𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)}) → (𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4745, 26, 46syl2anc 587 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
48 ovex 7173 . . . . . . 7 (𝐴m 𝐵) ∈ V
4948mptex 6968 . . . . . 6 (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∈ V
5049rnex 7606 . . . . 5 ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∈ V
51 ensym 8548 . . . . . . . . . . . 12 (𝑥𝐵𝐵𝑥)
5251ad2antll 728 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → 𝐵𝑥)
53 bren 8508 . . . . . . . . . . 11 (𝐵𝑥 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑥)
5452, 53sylib 221 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 𝑓:𝐵1-1-onto𝑥)
55 f1of 6594 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵𝑥)
5655adantl 485 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝑥)
57 simplrl 776 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥𝐴)
5856, 57fssd 6505 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝐴)
5911, 9elmapd 8410 . . . . . . . . . . . . . . 15 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
6059ad2antrr 725 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
6158, 60mpbird 260 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓 ∈ (𝐴m 𝐵))
62 f1ofo 6601 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵onto𝑥)
63 forn 6571 . . . . . . . . . . . . . . . 16 (𝑓:𝐵onto𝑥 → ran 𝑓 = 𝑥)
6462, 63syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝐵1-1-onto𝑥 → ran 𝑓 = 𝑥)
6564adantl 485 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → ran 𝑓 = 𝑥)
6665eqcomd 2804 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥 = ran 𝑓)
6761, 66jca 515 . . . . . . . . . . . 12 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓))
6867ex 416 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (𝑓:𝐵1-1-onto𝑥 → (𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓)))
6968eximdv 1918 . . . . . . . . . 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 3112 . . . . . . . . 9 (∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓))
7270, 71sylibr 237 . . . . . . . 8 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓)
7372ex 416 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥𝐴𝑥𝐵) → ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓))
7473ss2abdv 3991 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓})
75 eqid 2798 . . . . . . 7 (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓)
7675rnmpt 5792 . . . . . 6 ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓}
7774, 76sseqtrrdi 3966 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓))
78 ssdomg 8545 . . . . 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 3444 . . . . . . . . 9 𝑓 ∈ V
8180rnex 7606 . . . . . . . 8 ran 𝑓 ∈ V
8281rgenw 3118 . . . . . . 7 𝑓 ∈ (𝐴m 𝐵)ran 𝑓 ∈ V
8375fnmpt 6463 . . . . . . 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 6574 . . . . . 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 221 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓):(𝐴m 𝐵)–onto→ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓))
87 fodomnum 9475 . . . . 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 8552 . . . 4 (({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ≼ (𝐴m 𝐵)) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵))
9079, 88, 89syl2anc 587 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵))
91 sbth 8628 . . 3 (((𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ∧ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵)) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
9247, 90, 91syl2anc 587 . 2 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
937brrelex2i 5574 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94933ad2ant1 1130 . . . 4 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95 map0e 8436 . . . 4 (𝐴 ∈ V → (𝐴m ∅) = 1o)
9694, 95syl 17 . . 3 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m ∅) = 1o)
97 1oex 8100 . . . . 5 1o ∈ V
9897enref 8532 . . . 4 1o ≈ 1o
99 df-sn 4526 . . . . 5 {∅} = {𝑥𝑥 = ∅}
100 df1o2 8106 . . . . 5 1o = {∅}
101 en0 8562 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102101anbi2i 625 . . . . . . 7 ((𝑥𝐴𝑥 ≈ ∅) ↔ (𝑥𝐴𝑥 = ∅))
103 0ss 4304 . . . . . . . . 9 ∅ ⊆ 𝐴
104 sseq1 3940 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
105103, 104mpbiri 261 . . . . . . . 8 (𝑥 = ∅ → 𝑥𝐴)
106105pm4.71ri 564 . . . . . . 7 (𝑥 = ∅ ↔ (𝑥𝐴𝑥 = ∅))
107102, 106bitr4i 281 . . . . . 6 ((𝑥𝐴𝑥 ≈ ∅) ↔ 𝑥 = ∅)
108107abbii 2863 . . . . 5 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = {𝑥𝑥 = ∅}
10999, 100, 1083eqtr4ri 2832 . . . 4 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = 1o
11098, 109breqtrri 5058 . . 3 1o ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}
11196, 110eqbrtrdi 5070 . 2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
1125, 92, 111pm2.61ne 3072 1 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5031   ↦ cmpt 5111   × cxp 5518  dom cdm 5520  ran crn 5521  Fun wfun 6321   Fn wfn 6322  ⟶wf 6323  –onto→wfo 6325  –1-1-onto→wf1o 6326  (class class class)co 7140  ωcom 7567  1oc1o 8085   ↑m cmap 8396   ≈ cen 8496   ≼ cdom 8497  cardccrd 9355 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448  ax-inf2 9095 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4840  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-tr 5138  df-id 5426  df-eprel 5431  df-po 5439  df-so 5440  df-fr 5479  df-se 5480  df-we 5481  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-pred 6119  df-ord 6165  df-on 6166  df-lim 6167  df-suc 6168  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-isom 6336  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7568  df-1st 7678  df-2nd 7679  df-wrecs 7937  df-recs 7998  df-rdg 8036  df-1o 8092  df-oadd 8096  df-er 8279  df-map 8398  df-en 8500  df-dom 8501  df-sdom 8502  df-fin 8503  df-oi 8965  df-card 9359  df-acn 9362 This theorem is referenced by:  infmap  9994
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