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Theorem domssex 8666
 Description: Weakening of domssex 8666 to forget the functions in favor of dominance and equinumerosity. (Contributed by Mario Carneiro, 7-Feb-2015.) (Revised by Mario Carneiro, 24-Jun-2015.)
Assertion
Ref Expression
domssex (𝐴𝐵 → ∃𝑥(𝐴𝑥𝐵𝑥))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem domssex
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8507 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 reldom 8502 . . 3 Rel ≼
32brrelex2i 5586 . 2 (𝐴𝐵𝐵 ∈ V)
4 vex 3472 . . . . . . . 8 𝑓 ∈ V
5 f1stres 7699 . . . . . . . . 9 (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6 difexg 5207 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
76adantl 485 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8 snex 5309 . . . . . . . . . 10 {𝒫 ran 𝐴} ∈ V
9 xpexg 7458 . . . . . . . . . 10 (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
107, 8, 9sylancl 589 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
11 fex2 7624 . . . . . . . . 9 (((1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓) ∧ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V ∧ (𝐵 ∖ ran 𝑓) ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V)
125, 10, 7, 11mp3an2i 1463 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V)
13 unexg 7457 . . . . . . . 8 ((𝑓 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
144, 12, 13sylancr 590 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
15 cnvexg 7615 . . . . . . 7 ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
1614, 15syl 17 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
17 rnexg 7600 . . . . . 6 ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V → ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
1816, 17syl 17 . . . . 5 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
19 simpl 486 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝑓:𝐴1-1𝐵)
20 f1dm 6560 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 → dom 𝑓 = 𝐴)
214dmex 7602 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21eqeltrrdi 2923 . . . . . . . . 9 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
2322adantr 484 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ∈ V)
24 simpr 488 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ∈ V)
25 eqid 2822 . . . . . . . . 9 (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) = (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))
2625domss2 8664 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐴 ∈ V ∧ 𝐵 ∈ V) → ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∘ 𝑓) = ( I ↾ 𝐴)))
2719, 23, 24, 26syl3anc 1368 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ ((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∘ 𝑓) = ( I ↾ 𝐴)))
2827simp2d 1140 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
2927simp1d 1139 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
30 f1oen3g 8512 . . . . . . 7 (((𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V ∧ (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))):𝐵1-1-onto→ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))) → 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3116, 29, 30syl2anc 587 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3228, 31jca 515 . . . . 5 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
33 sseq2 3968 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐴𝑥𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
34 breq2 5046 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐵𝑥𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
3533, 34anbi12d 633 . . . . 5 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → ((𝐴𝑥𝐵𝑥) ↔ (𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∧ 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))))
3618, 32, 35spcedv 3574 . . . 4 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ∃𝑥(𝐴𝑥𝐵𝑥))
3736ex 416 . . 3 (𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
3837exlimiv 1931 . 2 (∃𝑓 𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
391, 3, 38sylc 65 1 (𝐴𝐵 → ∃𝑥(𝐴𝑥𝐵𝑥))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2114  Vcvv 3469   ∖ cdif 3905   ∪ cun 3906   ⊆ wss 3908  𝒫 cpw 4511  {csn 4539  ∪ cuni 4813   class class class wbr 5042   I cid 5436   × cxp 5530  ◡ccnv 5531  dom cdm 5532  ran crn 5533   ↾ cres 5534   ∘ ccom 5536  ⟶wf 6330  –1-1→wf1 6331  –1-1-onto→wf1o 6333  1st c1st 7673   ≈ cen 8493   ≼ cdom 8494 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 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-int 4852  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-1st 7675  df-2nd 7676  df-en 8497  df-dom 8498 This theorem is referenced by: (None)
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