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Theorem infmap2 10255
Description: An exponentiation law for infinite cardinals. Similar to Lemma 6.2 of [Jech] p. 43. Although this version of infmap 10614 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 7439 . . 3 (𝐵 = ∅ → (𝐴m 𝐵) = (𝐴m ∅))
2 breq2 5152 . . . . 5 (𝐵 = ∅ → (𝑥𝐵𝑥 ≈ ∅))
32anbi2d 630 . . . 4 (𝐵 = ∅ → ((𝑥𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥 ≈ ∅)))
43abbidv 2806 . . 3 (𝐵 = ∅ → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} = {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
51, 4breq12d 5161 . 2 (𝐵 = ∅ → ((𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ↔ (𝐴m ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}))
6 simpl2 1191 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵𝐴)
7 reldom 8990 . . . . . . . . . . 11 Rel ≼
87brrelex1i 5745 . . . . . . . . . 10 (𝐵𝐴𝐵 ∈ V)
96, 8syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ∈ V)
107brrelex2i 5746 . . . . . . . . . 10 (𝐵𝐴𝐴 ∈ V)
116, 10syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ V)
12 xpcomeng 9103 . . . . . . . . 9 ((𝐵 ∈ V ∧ 𝐴 ∈ V) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
139, 11, 12syl2anc 584 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ (𝐴 × 𝐵))
14 simpl3 1192 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ∈ dom card)
15 simpr 484 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐵 ≠ ∅)
16 mapdom3 9188 . . . . . . . . . . 11 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
1711, 9, 15, 16syl3anc 1370 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ≼ (𝐴m 𝐵))
18 numdom 10076 . . . . . . . . . 10 (((𝐴m 𝐵) ∈ dom card ∧ 𝐴 ≼ (𝐴m 𝐵)) → 𝐴 ∈ dom card)
1914, 17, 18syl2anc 584 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → 𝐴 ∈ dom card)
20 simpl1 1190 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ω ≼ 𝐴)
21 infxpabs 10249 . . . . . . . . 9 (((𝐴 ∈ dom card ∧ ω ≼ 𝐴) ∧ (𝐵 ≠ ∅ ∧ 𝐵𝐴)) → (𝐴 × 𝐵) ≈ 𝐴)
2219, 20, 15, 6, 21syl22anc 839 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴 × 𝐵) ≈ 𝐴)
23 entr 9045 . . . . . . . 8 (((𝐵 × 𝐴) ≈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ≈ 𝐴) → (𝐵 × 𝐴) ≈ 𝐴)
2413, 22, 23syl2anc 584 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐵 × 𝐴) ≈ 𝐴)
25 ssenen 9190 . . . . . . 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 8989 . . . . . . 7 Rel ≈
2827brrelex1i 5745 . . . . . 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 2877 . . . . . 6 {𝑥𝑥 ∈ (𝐴m 𝐵)} = (𝐴m 𝐵)
31 elmapi 8888 . . . . . . . 8 (𝑥 ∈ (𝐴m 𝐵) → 𝑥:𝐵𝐴)
32 fssxp 6764 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥 ⊆ (𝐵 × 𝐴))
33 ffun 6740 . . . . . . . . . . 11 (𝑥:𝐵𝐴 → Fun 𝑥)
34 vex 3482 . . . . . . . . . . . 12 𝑥 ∈ V
3534fundmen 9070 . . . . . . . . . . 11 (Fun 𝑥 → dom 𝑥𝑥)
36 ensym 9042 . . . . . . . . . . 11 (dom 𝑥𝑥𝑥 ≈ dom 𝑥)
3733, 35, 363syl 18 . . . . . . . . . 10 (𝑥:𝐵𝐴𝑥 ≈ dom 𝑥)
38 fdm 6746 . . . . . . . . . 10 (𝑥:𝐵𝐴 → dom 𝑥 = 𝐵)
3937, 38breqtrd 5174 . . . . . . . . 9 (𝑥:𝐵𝐴𝑥𝐵)
4032, 39jca 511 . . . . . . . 8 (𝑥:𝐵𝐴 → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4131, 40syl 17 . . . . . . 7 (𝑥 ∈ (𝐴m 𝐵) → (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵))
4241ss2abi 4077 . . . . . 6 {𝑥𝑥 ∈ (𝐴m 𝐵)} ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
4330, 42eqsstrri 4031 . . . . 5 (𝐴m 𝐵) ⊆ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)}
44 ssdomg 9039 . . . . 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 9052 . . . 4 (((𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ∧ {𝑥 ∣ (𝑥 ⊆ (𝐵 × 𝐴) ∧ 𝑥𝐵)} ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)}) → (𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
4745, 26, 46syl2anc 584 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
48 ovex 7464 . . . . . . 7 (𝐴m 𝐵) ∈ V
4948mptex 7243 . . . . . 6 (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∈ V
5049rnex 7933 . . . . 5 ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∈ V
51 ensym 9042 . . . . . . . . . . . 12 (𝑥𝐵𝐵𝑥)
5251ad2antll 729 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → 𝐵𝑥)
53 bren 8994 . . . . . . . . . . 11 (𝐵𝑥 ↔ ∃𝑓 𝑓:𝐵1-1-onto𝑥)
5452, 53sylib 218 . . . . . . . . . 10 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 𝑓:𝐵1-1-onto𝑥)
55 f1of 6849 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵𝑥)
5655adantl 481 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝑥)
57 simplrl 777 . . . . . . . . . . . . . . 15 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥𝐴)
5856, 57fssd 6754 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓:𝐵𝐴)
5911, 9elmapd 8879 . . . . . . . . . . . . . . 15 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
6059ad2antrr 726 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴m 𝐵) ↔ 𝑓:𝐵𝐴))
6158, 60mpbird 257 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑓 ∈ (𝐴m 𝐵))
62 f1ofo 6856 . . . . . . . . . . . . . . . 16 (𝑓:𝐵1-1-onto𝑥𝑓:𝐵onto𝑥)
63 forn 6824 . . . . . . . . . . . . . . . 16 (𝑓:𝐵onto𝑥 → ran 𝑓 = 𝑥)
6462, 63syl 17 . . . . . . . . . . . . . . 15 (𝑓:𝐵1-1-onto𝑥 → ran 𝑓 = 𝑥)
6564adantl 481 . . . . . . . . . . . . . 14 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → ran 𝑓 = 𝑥)
6665eqcomd 2741 . . . . . . . . . . . . 13 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → 𝑥 = ran 𝑓)
6761, 66jca 511 . . . . . . . . . . . 12 (((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) ∧ 𝑓:𝐵1-1-onto𝑥) → (𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓))
6867ex 412 . . . . . . . . . . 11 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → (𝑓:𝐵1-1-onto𝑥 → (𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓)))
6968eximdv 1915 . . . . . . . . . 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 3069 . . . . . . . . 9 (∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓 ↔ ∃𝑓(𝑓 ∈ (𝐴m 𝐵) ∧ 𝑥 = ran 𝑓))
7270, 71sylibr 234 . . . . . . . 8 ((((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) ∧ (𝑥𝐴𝑥𝐵)) → ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓)
7372ex 412 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → ((𝑥𝐴𝑥𝐵) → ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓))
7473ss2abdv 4076 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ {𝑥 ∣ ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓})
75 eqid 2735 . . . . . . 7 (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) = (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓)
7675rnmpt 5971 . . . . . 6 ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) = {𝑥 ∣ ∃𝑓 ∈ (𝐴m 𝐵)𝑥 = ran 𝑓}
7774, 76sseqtrrdi 4047 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ⊆ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓))
78 ssdomg 9039 . . . . 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 3482 . . . . . . . . 9 𝑓 ∈ V
8180rnex 7933 . . . . . . . 8 ran 𝑓 ∈ V
8281rgenw 3063 . . . . . . 7 𝑓 ∈ (𝐴m 𝐵)ran 𝑓 ∈ V
8375fnmpt 6709 . . . . . . 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 6827 . . . . . 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 218 . . . . 5 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓):(𝐴m 𝐵)–onto→ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓))
87 fodomnum 10095 . . . . 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 9046 . . . 4 (({𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ∧ ran (𝑓 ∈ (𝐴m 𝐵) ↦ ran 𝑓) ≼ (𝐴m 𝐵)) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵))
9079, 88, 89syl2anc 584 . . 3 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵))
91 sbth 9132 . . 3 (((𝐴m 𝐵) ≼ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ∧ {𝑥 ∣ (𝑥𝐴𝑥𝐵)} ≼ (𝐴m 𝐵)) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
9247, 90, 91syl2anc 584 . 2 (((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) ∧ 𝐵 ≠ ∅) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
937brrelex2i 5746 . . . . 5 (ω ≼ 𝐴𝐴 ∈ V)
94933ad2ant1 1132 . . . 4 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → 𝐴 ∈ V)
95 map0e 8921 . . . 4 (𝐴 ∈ V → (𝐴m ∅) = 1o)
9694, 95syl 17 . . 3 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m ∅) = 1o)
97 1oex 8515 . . . . 5 1o ∈ V
9897enref 9024 . . . 4 1o ≈ 1o
99 df-sn 4632 . . . . 5 {∅} = {𝑥𝑥 = ∅}
100 df1o2 8512 . . . . 5 1o = {∅}
101 en0 9057 . . . . . . . 8 (𝑥 ≈ ∅ ↔ 𝑥 = ∅)
102101anbi2i 623 . . . . . . 7 ((𝑥𝐴𝑥 ≈ ∅) ↔ (𝑥𝐴𝑥 = ∅))
103 0ss 4406 . . . . . . . . 9 ∅ ⊆ 𝐴
104 sseq1 4021 . . . . . . . . 9 (𝑥 = ∅ → (𝑥𝐴 ↔ ∅ ⊆ 𝐴))
105103, 104mpbiri 258 . . . . . . . 8 (𝑥 = ∅ → 𝑥𝐴)
106105pm4.71ri 560 . . . . . . 7 (𝑥 = ∅ ↔ (𝑥𝐴𝑥 = ∅))
107102, 106bitr4i 278 . . . . . 6 ((𝑥𝐴𝑥 ≈ ∅) ↔ 𝑥 = ∅)
108107abbii 2807 . . . . 5 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = {𝑥𝑥 = ∅}
10999, 100, 1083eqtr4ri 2774 . . . 4 {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)} = 1o
11098, 109breqtrri 5175 . . 3 1o ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)}
11196, 110eqbrtrdi 5187 . 2 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m ∅) ≈ {𝑥 ∣ (𝑥𝐴𝑥 ≈ ∅)})
1125, 92, 111pm2.61ne 3025 1 ((ω ≼ 𝐴𝐵𝐴 ∧ (𝐴m 𝐵) ∈ dom card) → (𝐴m 𝐵) ≈ {𝑥 ∣ (𝑥𝐴𝑥𝐵)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wne 2938  wral 3059  wrex 3068  Vcvv 3478  wss 3963  c0 4339  {csn 4631   class class class wbr 5148  cmpt 5231   × cxp 5687  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  wf 6559  ontowfo 6561  1-1-ontowf1o 6562  (class class class)co 7431  ωcom 7887  1oc1o 8498  m cmap 8865  cen 8981  cdom 8982  cardccrd 9973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-inf2 9679
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-oi 9548  df-card 9977  df-acn 9980
This theorem is referenced by:  infmap  10614
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