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Theorem 1stpreima 32531
Description: The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
1stpreima (š“ āŠ† šµ → (ā—”(1st ↾ (šµ Ɨ š¶)) ā€œ š“) = (š“ Ɨ š¶))

Proof of Theorem 1stpreima
Dummy variable š‘¤ is distinct from all other variables.
StepHypRef Expression
1 elxp7 8026 . . . . . 6 (š‘¤ ∈ (šµ Ɨ š¶) ↔ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (2nd ā€˜š‘¤) ∈ š¶)))
21anbi2i 621 . . . . 5 (((1st ā€˜š‘¤) ∈ š“ ∧ š‘¤ ∈ (šµ Ɨ š¶)) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (2nd ā€˜š‘¤) ∈ š¶))))
3 anass 467 . . . . . . 7 ((((1st ā€˜š‘¤) ∈ š“ ∧ (1st ā€˜š‘¤) ∈ šµ) ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶))))
43a1i 11 . . . . . 6 (š“ āŠ† šµ → ((((1st ā€˜š‘¤) ∈ š“ ∧ (1st ā€˜š‘¤) ∈ šµ) ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)))))
5 ssel 3965 . . . . . . . 8 (š“ āŠ† šµ → ((1st ā€˜š‘¤) ∈ š“ → (1st ā€˜š‘¤) ∈ šµ))
65pm4.71d 560 . . . . . . 7 (š“ āŠ† šµ → ((1st ā€˜š‘¤) ∈ š“ ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ (1st ā€˜š‘¤) ∈ šµ)))
76anbi1d 629 . . . . . 6 (š“ āŠ† šµ → (((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)) ↔ (((1st ā€˜š‘¤) ∈ š“ ∧ (1st ā€˜š‘¤) ∈ šµ) ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶))))
8 an12 643 . . . . . . . 8 ((š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (2nd ā€˜š‘¤) ∈ š¶)) ↔ ((1st ā€˜š‘¤) ∈ šµ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)))
98anbi2i 621 . . . . . . 7 (((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (2nd ā€˜š‘¤) ∈ š¶))) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶))))
109a1i 11 . . . . . 6 (š“ āŠ† šµ → (((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (2nd ā€˜š‘¤) ∈ š¶))) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)))))
114, 7, 103bitr4d 310 . . . . 5 (š“ āŠ† šµ → (((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ šµ ∧ (2nd ā€˜š‘¤) ∈ š¶)))))
122, 11bitr4id 289 . . . 4 (š“ āŠ† šµ → (((1st ā€˜š‘¤) ∈ š“ ∧ š‘¤ ∈ (šµ Ɨ š¶)) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶))))
13 an12 643 . . . 4 (((1st ā€˜š‘¤) ∈ š“ ∧ (š‘¤ ∈ (V Ɨ V) ∧ (2nd ā€˜š‘¤) ∈ š¶)) ↔ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ š“ ∧ (2nd ā€˜š‘¤) ∈ š¶)))
1412, 13bitrdi 286 . . 3 (š“ āŠ† šµ → (((1st ā€˜š‘¤) ∈ š“ ∧ š‘¤ ∈ (šµ Ɨ š¶)) ↔ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ š“ ∧ (2nd ā€˜š‘¤) ∈ š¶))))
15 cnvresima 6229 . . . . 5 (ā—”(1st ↾ (šµ Ɨ š¶)) ā€œ š“) = ((ā—”1st ā€œ š“) ∩ (šµ Ɨ š¶))
1615eleq2i 2817 . . . 4 (š‘¤ ∈ (ā—”(1st ↾ (šµ Ɨ š¶)) ā€œ š“) ↔ š‘¤ ∈ ((ā—”1st ā€œ š“) ∩ (šµ Ɨ š¶)))
17 elin 3955 . . . 4 (š‘¤ ∈ ((ā—”1st ā€œ š“) ∩ (šµ Ɨ š¶)) ↔ (š‘¤ ∈ (ā—”1st ā€œ š“) ∧ š‘¤ ∈ (šµ Ɨ š¶)))
18 vex 3467 . . . . . 6 š‘¤ ∈ V
19 fo1st 8011 . . . . . . 7 1st :V–onto→V
20 fofn 6808 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
21 elpreima 7062 . . . . . . 7 (1st Fn V → (š‘¤ ∈ (ā—”1st ā€œ š“) ↔ (š‘¤ ∈ V ∧ (1st ā€˜š‘¤) ∈ š“)))
2219, 20, 21mp2b 10 . . . . . 6 (š‘¤ ∈ (ā—”1st ā€œ š“) ↔ (š‘¤ ∈ V ∧ (1st ā€˜š‘¤) ∈ š“))
2318, 22mpbiran 707 . . . . 5 (š‘¤ ∈ (ā—”1st ā€œ š“) ↔ (1st ā€˜š‘¤) ∈ š“)
2423anbi1i 622 . . . 4 ((š‘¤ ∈ (ā—”1st ā€œ š“) ∧ š‘¤ ∈ (šµ Ɨ š¶)) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ š‘¤ ∈ (šµ Ɨ š¶)))
2516, 17, 243bitri 296 . . 3 (š‘¤ ∈ (ā—”(1st ↾ (šµ Ɨ š¶)) ā€œ š“) ↔ ((1st ā€˜š‘¤) ∈ š“ ∧ š‘¤ ∈ (šµ Ɨ š¶)))
26 elxp7 8026 . . 3 (š‘¤ ∈ (š“ Ɨ š¶) ↔ (š‘¤ ∈ (V Ɨ V) ∧ ((1st ā€˜š‘¤) ∈ š“ ∧ (2nd ā€˜š‘¤) ∈ š¶)))
2714, 25, 263bitr4g 313 . 2 (š“ āŠ† šµ → (š‘¤ ∈ (ā—”(1st ↾ (šµ Ɨ š¶)) ā€œ š“) ↔ š‘¤ ∈ (š“ Ɨ š¶)))
2827eqrdv 2723 1 (š“ āŠ† šµ → (ā—”(1st ↾ (šµ Ɨ š¶)) ā€œ š“) = (š“ Ɨ š¶))
Colors of variables: wff setvar class
Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1533   ∈ wcel 2098  Vcvv 3463   ∩ cin 3938   āŠ† wss 3939   Ɨ cxp 5670  ā—”ccnv 5671   ↾ cres 5674   ā€œ cima 5675   Fn wfn 6538  ā€“onto→wfo 6541  ā€˜cfv 6543  1st c1st 7989  2nd c2nd 7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7991  df-2nd 7992
This theorem is referenced by:  sxbrsigalem2  33963
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