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Theorem 1stpreima 30466
Description: The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
1stpreima (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Proof of Theorem 1stpreima
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 anass 472 . . . . . . 7 ((((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
21a1i 11 . . . . . 6 (𝐴𝐵 → ((((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))))
3 ssel 3908 . . . . . . . 8 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 → (1st𝑤) ∈ 𝐵))
43pm4.71d 565 . . . . . . 7 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 ↔ ((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵)))
54anbi1d 632 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
6 an12 644 . . . . . . . 8 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))
76anbi2i 625 . . . . . . 7 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
87a1i 11 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))))
92, 5, 83bitr4d 314 . . . . 5 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))))
10 elxp7 7706 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1110anbi2i 625 . . . . 5 (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))))
129, 11syl6rbbr 293 . . . 4 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
13 an12 644 . . . 4 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
1412, 13syl6bb 290 . . 3 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶))))
15 cnvresima 6054 . . . . 5 ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((1st𝐴) ∩ (𝐵 × 𝐶))
1615eleq2i 2881 . . . 4 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)))
17 elin 3897 . . . 4 (𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18 vex 3444 . . . . . 6 𝑤 ∈ V
19 fo1st 7691 . . . . . . 7 1st :V–onto→V
20 fofn 6567 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
21 elpreima 6805 . . . . . . 7 (1st Fn V → (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴)))
2219, 20, 21mp2b 10 . . . . . 6 (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴))
2318, 22mpbiran 708 . . . . 5 (𝑤 ∈ (1st𝐴) ↔ (1st𝑤) ∈ 𝐴)
2423anbi1i 626 . . . 4 ((𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2516, 17, 243bitri 300 . . 3 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
26 elxp7 7706 . . 3 (𝑤 ∈ (𝐴 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
2714, 25, 263bitr4g 317 . 2 (𝐴𝐵 → (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
2827eqrdv 2796 1 (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  Vcvv 3441  cin 3880  wss 3881   × cxp 5517  ccnv 5518  cres 5521  cima 5522   Fn wfn 6319  ontowfo 6322  cfv 6324  1st c1st 7669  2nd c2nd 7670
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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fo 6330  df-fv 6332  df-1st 7671  df-2nd 7672
This theorem is referenced by:  sxbrsigalem2  31654
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