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Theorem 1stconst 8032
Description: The mapping of a restriction of the 1st function to a constant function. (Contributed by NM, 14-Dec-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
1stconst (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)

Proof of Theorem 1stconst
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snnzg 4735 . . 3 (𝐵𝑉 → {𝐵} ≠ ∅)
2 fo1stres 7947 . . 3 ({𝐵} ≠ ∅ → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
31, 2syl 17 . 2 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴)
4 moeq 3665 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝑦, 𝐵
54moani 2551 . . . . 5 ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)
6 vex 3449 . . . . . . . 8 𝑦 ∈ V
76brresi 5946 . . . . . . 7 (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
8 fo1st 7941 . . . . . . . . . . 11 1st :V–onto→V
9 fofn 6758 . . . . . . . . . . 11 (1st :V–onto→V → 1st Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 1st Fn V
11 vex 3449 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6895 . . . . . . . . . 10 ((1st Fn V ∧ 𝑥 ∈ V) → ((1st𝑥) = 𝑦𝑥1st 𝑦))
1310, 11, 12mp2an 690 . . . . . . . . 9 ((1st𝑥) = 𝑦𝑥1st 𝑦)
1413anbi2i 623 . . . . . . . 8 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦))
15 elxp7 7956 . . . . . . . . . . 11 (𝑥 ∈ (𝐴 × {𝐵}) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})))
16 eleq1 2825 . . . . . . . . . . . . . . 15 ((1st𝑥) = 𝑦 → ((1st𝑥) ∈ 𝐴𝑦𝐴))
1716biimpac 479 . . . . . . . . . . . . . 14 (((1st𝑥) ∈ 𝐴 ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1817adantlr 713 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
1918adantll 712 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑦𝐴)
20 elsni 4603 . . . . . . . . . . . . . 14 ((2nd𝑥) ∈ {𝐵} → (2nd𝑥) = 𝐵)
21 eqopi 7957 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝑦 ∧ (2nd𝑥) = 𝐵)) → 𝑥 = ⟨𝑦, 𝐵⟩)
2221anass1rs 653 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) = 𝐵) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2320, 22sylanl2 679 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (2nd𝑥) ∈ {𝐵}) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2423adantlrl 718 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → 𝑥 = ⟨𝑦, 𝐵⟩)
2519, 24jca 512 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ {𝐵})) ∧ (1st𝑥) = 𝑦) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2615, 25sylanb 581 . . . . . . . . . 10 ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
2726adantl 482 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦)) → (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩))
28 simprr 771 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 = ⟨𝑦, 𝐵⟩)
29 simprl 769 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑦𝐴)
30 snidg 4620 . . . . . . . . . . . . 13 (𝐵𝑉𝐵 ∈ {𝐵})
3130adantr 481 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵 ∈ {𝐵})
3229, 31opelxpd 5671 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → ⟨𝑦, 𝐵⟩ ∈ (𝐴 × {𝐵}))
3328, 32eqeltrd 2838 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝑥 ∈ (𝐴 × {𝐵}))
3428fveq2d 6846 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = (1st ‘⟨𝑦, 𝐵⟩))
35 simpl 483 . . . . . . . . . . . 12 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → 𝐵𝑉)
36 op1stg 7933 . . . . . . . . . . . 12 ((𝑦𝐴𝐵𝑉) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3729, 35, 36syl2anc 584 . . . . . . . . . . 11 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st ‘⟨𝑦, 𝐵⟩) = 𝑦)
3834, 37eqtrd 2776 . . . . . . . . . 10 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (1st𝑥) = 𝑦)
3933, 38jca 512 . . . . . . . . 9 ((𝐵𝑉 ∧ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)) → (𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦))
4027, 39impbida 799 . . . . . . . 8 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ (1st𝑥) = 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4114, 40bitr3id 284 . . . . . . 7 (𝐵𝑉 → ((𝑥 ∈ (𝐴 × {𝐵}) ∧ 𝑥1st 𝑦) ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
427, 41bitrid 282 . . . . . 6 (𝐵𝑉 → (𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ (𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
4342mobidv 2547 . . . . 5 (𝐵𝑉 → (∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦 ↔ ∃*𝑥(𝑦𝐴𝑥 = ⟨𝑦, 𝐵⟩)))
445, 43mpbiri 257 . . . 4 (𝐵𝑉 → ∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4544alrimiv 1930 . . 3 (𝐵𝑉 → ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
46 funcnv2 6569 . . 3 (Fun (1st ↾ (𝐴 × {𝐵})) ↔ ∀𝑦∃*𝑥 𝑥(1st ↾ (𝐴 × {𝐵}))𝑦)
4745, 46sylibr 233 . 2 (𝐵𝑉 → Fun (1st ↾ (𝐴 × {𝐵})))
48 dff1o3 6790 . 2 ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴 ↔ ((1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–onto𝐴 ∧ Fun (1st ↾ (𝐴 × {𝐵}))))
493, 47, 48sylanbrc 583 1 (𝐵𝑉 → (1st ↾ (𝐴 × {𝐵})):(𝐴 × {𝐵})–1-1-onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wal 1539   = wceq 1541  wcel 2106  ∃*wmo 2536  wne 2943  Vcvv 3445  c0 4282  {csn 4586  cop 4592   class class class wbr 5105   × cxp 5631  ccnv 5632  cres 5635  Fun wfun 6490   Fn wfn 6491  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496  1st c1st 7919  2nd c2nd 7920
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-1st 7921  df-2nd 7922
This theorem is referenced by:  curry2  8039  domss2  9080  fv1stcnv  34351
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