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Theorem domssex 9177
Description: Weakening of domssex2 9176 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 8998 . 2 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
2 reldom 8990 . . 3 Rel ≼
32brrelex2i 5746 . 2 (𝐴𝐵𝐵 ∈ V)
4 vex 3482 . . . . . . . 8 𝑓 ∈ V
5 f1stres 8037 . . . . . . . . 9 (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})):((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})⟶(𝐵 ∖ ran 𝑓)
6 difexg 5335 . . . . . . . . . . 11 (𝐵 ∈ V → (𝐵 ∖ ran 𝑓) ∈ V)
76adantl 481 . . . . . . . . . 10 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → (𝐵 ∖ ran 𝑓) ∈ V)
8 snex 5442 . . . . . . . . . 10 {𝒫 ran 𝐴} ∈ V
9 xpexg 7769 . . . . . . . . . 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 7957 . . . . . . . . 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 7762 . . . . . . . 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 7947 . . . . . . 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 7925 . . . . . 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 482 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝑓:𝐴1-1𝐵)
20 f1dm 6809 . . . . . . . . . 10 (𝑓:𝐴1-1𝐵 → dom 𝑓 = 𝐴)
214dmex 7932 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21eqeltrrdi 2848 . . . . . . . . 9 (𝑓:𝐴1-1𝐵𝐴 ∈ V)
2322adantr 480 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐴 ∈ V)
24 simpr 484 . . . . . . . 8 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → 𝐵 ∈ V)
25 eqid 2735 . . . . . . . . 9 (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) = (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))
2625domss2 9175 . . . . . . . 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 9006 . . . . . . 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 511 . . . . 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 4022 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐴𝑥𝐴 ⊆ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
34 breq2 5152 . . . . . 6 (𝑥 = ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴}))) → (𝐵𝑥𝐵 ≈ ran (𝑓 ∪ (1st ↾ ((𝐵 ∖ ran 𝑓) × {𝒫 ran 𝐴})))))
3533, 34anbi12d 632 . . . . 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 3598 . . . 4 ((𝑓:𝐴1-1𝐵𝐵 ∈ V) → ∃𝑥(𝐴𝑥𝐵𝑥))
3736ex 412 . . 3 (𝑓:𝐴1-1𝐵 → (𝐵 ∈ V → ∃𝑥(𝐴𝑥𝐵𝑥)))
3837exlimiv 1928 . 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 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478  cdif 3960  cun 3961  wss 3963  𝒫 cpw 4605  {csn 4631   cuni 4912   class class class wbr 5148   I cid 5582   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cres 5691  ccom 5693  wf 6559  1-1wf1 6560  1-1-ontowf1o 6562  1st c1st 8011  cen 8981  cdom 8982
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-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  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-id 5583  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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014  df-en 8985  df-dom 8986
This theorem is referenced by: (None)
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