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Theorem domssex 9140
Description: Weakening of domssex2 9139 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 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡
Proof of Theorem domssex
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8956 . 2 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡)
2 reldom 8947 . . 3 Rel β‰Ό
32brrelex2i 5732 . 2 (𝐴 β‰Ό 𝐡 β†’ 𝐡 ∈ V)
4 vex 3476 . . . . . . . 8 𝑓 ∈ V
5 f1stres 8001 . . . . . . . . 9 (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})⟢(𝐡 βˆ– ran 𝑓)
6 difexg 5326 . . . . . . . . . . 11 (𝐡 ∈ V β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
76adantl 480 . . . . . . . . . 10 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝐡 βˆ– ran 𝑓) ∈ V)
8 snex 5430 . . . . . . . . . 10 {𝒫 βˆͺ ran 𝐴} ∈ V
9 xpexg 7739 . . . . . . . . . 10 (((𝐡 βˆ– ran 𝑓) ∈ V ∧ {𝒫 βˆͺ ran 𝐴} ∈ V) β†’ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
107, 8, 9sylancl 584 . . . . . . . . 9 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V)
11 fex2 7926 . . . . . . . . 9 (((1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})):((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})⟢(𝐡 βˆ– ran 𝑓) ∧ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}) ∈ V ∧ (𝐡 βˆ– ran 𝑓) ∈ V) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
125, 10, 7, 11mp3an2i 1464 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V)
13 unexg 7738 . . . . . . . 8 ((𝑓 ∈ V ∧ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})) ∈ V) β†’ (𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
144, 12, 13sylancr 585 . . . . . . 7 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
15 cnvexg 7917 . . . . . . 7 ((𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V β†’ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
1614, 15syl 17 . . . . . 6 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
17 rnexg 7897 . . . . . 6 (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V β†’ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
1816, 17syl 17 . . . . 5 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V)
19 simpl 481 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝑓:𝐴–1-1→𝐡)
20 f1dm 6790 . . . . . . . . . 10 (𝑓:𝐴–1-1→𝐡 β†’ dom 𝑓 = 𝐴)
214dmex 7904 . . . . . . . . . 10 dom 𝑓 ∈ V
2220, 21eqeltrrdi 2840 . . . . . . . . 9 (𝑓:𝐴–1-1→𝐡 β†’ 𝐴 ∈ V)
2322adantr 479 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 ∈ V)
24 simpr 483 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐡 ∈ V)
25 eqid 2730 . . . . . . . . 9 β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) = β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))
2625domss2 9138 . . . . . . . 8 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐴 ∈ V ∧ 𝐡 ∈ V) β†’ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝑓) = ( I β†Ύ 𝐴)))
2719, 23, 24, 26syl3anc 1369 . . . . . . 7 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ (β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∘ 𝑓) = ( I β†Ύ 𝐴)))
2827simp2d 1141 . . . . . 6 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
2927simp1d 1140 . . . . . . 7 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
30 f1oen3g 8964 . . . . . . 7 ((β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∈ V ∧ β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))):𝐡–1-1-ontoβ†’ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))) β†’ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
3116, 29, 30syl2anc 582 . . . . . 6 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))
3228, 31jca 510 . . . . 5 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ (𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))))
33 sseq2 4007 . . . . . 6 (π‘₯ = ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝐴 βŠ† π‘₯ ↔ 𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))))
34 breq2 5151 . . . . . 6 (π‘₯ = ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ (𝐡 β‰ˆ π‘₯ ↔ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴})))))
3533, 34anbi12d 629 . . . . 5 (π‘₯ = ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) β†’ ((𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯) ↔ (𝐴 βŠ† ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))) ∧ 𝐡 β‰ˆ ran β—‘(𝑓 βˆͺ (1st β†Ύ ((𝐡 βˆ– ran 𝑓) Γ— {𝒫 βˆͺ ran 𝐴}))))))
3618, 32, 35spcedv 3587 . . . 4 ((𝑓:𝐴–1-1→𝐡 ∧ 𝐡 ∈ V) β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
3736ex 411 . . 3 (𝑓:𝐴–1-1→𝐡 β†’ (𝐡 ∈ V β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯)))
3837exlimiv 1931 . 2 (βˆƒπ‘“ 𝑓:𝐴–1-1→𝐡 β†’ (𝐡 ∈ V β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯)))
391, 3, 38sylc 65 1 (𝐴 β‰Ό 𝐡 β†’ βˆƒπ‘₯(𝐴 βŠ† π‘₯ ∧ 𝐡 β‰ˆ π‘₯))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104  Vcvv 3472   βˆ– cdif 3944   βˆͺ cun 3945   βŠ† wss 3947  π’« cpw 4601  {csn 4627  βˆͺ cuni 4907   class class class wbr 5147   I cid 5572   Γ— cxp 5673  β—‘ccnv 5674  dom cdm 5675  ran crn 5676   β†Ύ cres 5677   ∘ ccom 5679  βŸΆwf 6538  β€“1-1β†’wf1 6539  β€“1-1-ontoβ†’wf1o 6541  1st c1st 7975   β‰ˆ cen 8938   β‰Ό cdom 8939
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1st 7977  df-2nd 7978  df-en 8942  df-dom 8943
This theorem is referenced by: (None)
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