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Theorem domssex 8905
Description: Weakening of domssex2 8904 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 8729 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 reldom 8720 . . 3 Rel ≼
32brrelex2i 5644 . 2 (𝐴𝐵𝐵 ∈ V)
4 vex 3435 . . . . . . . 8 𝑓 ∈ V
5 f1stres 7846 . . . . . . . . 9 (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6 difexg 5255 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
76adantl 482 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8 snex 5358 . . . . . . . . . 10 {𝒫 ran 𝐴} ∈ V
9 xpexg 7592 . . . . . . . . . 10 (((𝐵 ∖ ran 𝑓) ∈ V ∧ {𝒫 ran 𝐴} ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
107, 8, 9sylancl 586 . . . . . . . . 9 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}) ∈ V)
11 fex2 7772 . . . . . . . . 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 1465 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V)
13 unexg 7591 . . . . . . . 8 ((𝑓 ∈ V ∧ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})) ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
144, 12, 13sylancr 587 . . . . . . 7 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) ∈ V)
15 cnvexg 7763 . . . . . . 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 7743 . . . . . 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 483 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝑓:𝐴1-1𝐵)
20 f1dm 6671 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 → dom 𝑓 = 𝐴)
214dmex 7750 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21eqeltrrdi 2850 . . . . . . . . 9 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
2322adantr 481 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ∈ V)
24 simpr 485 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ∈ V)
25 eqid 2740 . . . . . . . . 9 (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) = (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))
2625domss2 8903 . . . . . . . 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 1370 . . . . . . 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 1142 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
2927simp1d 1141 . . . . . . 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 8735 . . . . . . 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 584 . . . . . 6 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))))
3228, 31jca 512 . . . . 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 3952 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐴𝑥𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
34 breq2 5083 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐵𝑥𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
3533, 34anbi12d 631 . . . . 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 3536 . . . 4 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ∃𝑥(𝐴𝑥𝐵𝑥))
3736ex 413 . . 3 (𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
3837exlimiv 1937 . 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 396  w3a 1086   = wceq 1542  wex 1786  wcel 2110  Vcvv 3431  cdif 3889  cun 3890  wss 3892  𝒫 cpw 4539  {csn 4567   cuni 4845   class class class wbr 5079   I cid 5488   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cres 5591  ccom 5593  wf 6427  1-1wf1 6428  1-1-ontowf1o 6430  1st c1st 7820  cen 8711  cdom 8712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-1st 7822  df-2nd 7823  df-en 8715  df-dom 8716
This theorem is referenced by: (None)
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