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Theorem fmucndlem 22503
Description: Lemma for fmucnd 22504. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
fmucndlem ((𝐹 Fn 𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹𝐴) × (𝐹𝐴)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑋,𝑦

Proof of Theorem fmucndlem
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 df-ima 5368 . . 3 ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝐴 × 𝐴)) = ran ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↾ (𝐴 × 𝐴))
2 simpr 479 . . . . 5 ((𝐹 Fn 𝑋𝐴𝑋) → 𝐴𝑋)
3 resmpt2 7035 . . . . 5 ((𝐴𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↾ (𝐴 × 𝐴)) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩))
42, 3sylancom 582 . . . 4 ((𝐹 Fn 𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↾ (𝐴 × 𝐴)) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩))
54rneqd 5598 . . 3 ((𝐹 Fn 𝑋𝐴𝑋) → ran ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↾ (𝐴 × 𝐴)) = ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩))
61, 5syl5eq 2826 . 2 ((𝐹 Fn 𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝐴 × 𝐴)) = ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩))
7 vex 3401 . . . . . . . . . . . . 13 𝑥 ∈ V
8 vex 3401 . . . . . . . . . . . . 13 𝑦 ∈ V
97, 8op1std 7455 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (1st𝑝) = 𝑥)
109fveq2d 6450 . . . . . . . . . . 11 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(1st𝑝)) = (𝐹𝑥))
117, 8op2ndd 7456 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (2nd𝑝) = 𝑦)
1211fveq2d 6450 . . . . . . . . . . 11 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝐹‘(2nd𝑝)) = (𝐹𝑦))
1310, 12opeq12d 4644 . . . . . . . . . 10 (𝑝 = ⟨𝑥, 𝑦⟩ → ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
1413mpt2mpt 7029 . . . . . . . . 9 (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
1514eqcomi 2787 . . . . . . . 8 (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
1615rneqi 5597 . . . . . . 7 ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = ran (𝑝 ∈ (𝐴 × 𝐴) ↦ ⟨(𝐹‘(1st𝑝)), (𝐹‘(2nd𝑝))⟩)
17 fvexd 6461 . . . . . . 7 ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(1st𝑝)) ∈ V)
18 fvexd 6461 . . . . . . 7 ((⊤ ∧ 𝑝 ∈ (𝐴 × 𝐴)) → (𝐹‘(2nd𝑝)) ∈ V)
1916, 17, 18fliftrel 6830 . . . . . 6 (⊤ → ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ⊆ (V × V))
2019mptru 1609 . . . . 5 ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ⊆ (V × V)
2120sseli 3817 . . . 4 (𝑝 ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) → 𝑝 ∈ (V × V))
2221adantl 475 . . 3 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)) → 𝑝 ∈ (V × V))
23 xpss 5371 . . . . 5 ((𝐹𝐴) × (𝐹𝐴)) ⊆ (V × V)
2423sseli 3817 . . . 4 (𝑝 ∈ ((𝐹𝐴) × (𝐹𝐴)) → 𝑝 ∈ (V × V))
2524adantl 475 . . 3 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ ((𝐹𝐴) × (𝐹𝐴))) → 𝑝 ∈ (V × V))
26 fvelimab 6513 . . . . . . . 8 ((𝐹 Fn 𝑋𝐴𝑋) → ((1st𝑝) ∈ (𝐹𝐴) ↔ ∃𝑥𝐴 (𝐹𝑥) = (1st𝑝)))
27 fvelimab 6513 . . . . . . . 8 ((𝐹 Fn 𝑋𝐴𝑋) → ((2nd𝑝) ∈ (𝐹𝐴) ↔ ∃𝑦𝐴 (𝐹𝑦) = (2nd𝑝)))
2826, 27anbi12d 624 . . . . . . 7 ((𝐹 Fn 𝑋𝐴𝑋) → (((1st𝑝) ∈ (𝐹𝐴) ∧ (2nd𝑝) ∈ (𝐹𝐴)) ↔ (∃𝑥𝐴 (𝐹𝑥) = (1st𝑝) ∧ ∃𝑦𝐴 (𝐹𝑦) = (2nd𝑝))))
29 eqid 2778 . . . . . . . . 9 (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)
30 opex 5164 . . . . . . . . 9 ⟨(𝐹𝑥), (𝐹𝑦)⟩ ∈ V
3129, 30elrnmpt2 7050 . . . . . . . 8 (⟨(1st𝑝), (2nd𝑝)⟩ ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ ∃𝑥𝐴𝑦𝐴 ⟨(1st𝑝), (2nd𝑝)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩)
32 eqcom 2785 . . . . . . . . . 10 (⟨(1st𝑝), (2nd𝑝)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩ ↔ ⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩)
33 fvex 6459 . . . . . . . . . . 11 (1st𝑝) ∈ V
34 fvex 6459 . . . . . . . . . . 11 (2nd𝑝) ∈ V
3533, 34opth2 5180 . . . . . . . . . 10 (⟨(𝐹𝑥), (𝐹𝑦)⟩ = ⟨(1st𝑝), (2nd𝑝)⟩ ↔ ((𝐹𝑥) = (1st𝑝) ∧ (𝐹𝑦) = (2nd𝑝)))
3632, 35bitri 267 . . . . . . . . 9 (⟨(1st𝑝), (2nd𝑝)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩ ↔ ((𝐹𝑥) = (1st𝑝) ∧ (𝐹𝑦) = (2nd𝑝)))
37362rexbii 3225 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ⟨(1st𝑝), (2nd𝑝)⟩ = ⟨(𝐹𝑥), (𝐹𝑦)⟩ ↔ ∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (1st𝑝) ∧ (𝐹𝑦) = (2nd𝑝)))
38 reeanv 3293 . . . . . . . 8 (∃𝑥𝐴𝑦𝐴 ((𝐹𝑥) = (1st𝑝) ∧ (𝐹𝑦) = (2nd𝑝)) ↔ (∃𝑥𝐴 (𝐹𝑥) = (1st𝑝) ∧ ∃𝑦𝐴 (𝐹𝑦) = (2nd𝑝)))
3931, 37, 383bitri 289 . . . . . . 7 (⟨(1st𝑝), (2nd𝑝)⟩ ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ (∃𝑥𝐴 (𝐹𝑥) = (1st𝑝) ∧ ∃𝑦𝐴 (𝐹𝑦) = (2nd𝑝)))
4028, 39syl6rbbr 282 . . . . . 6 ((𝐹 Fn 𝑋𝐴𝑋) → (⟨(1st𝑝), (2nd𝑝)⟩ ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ ((1st𝑝) ∈ (𝐹𝐴) ∧ (2nd𝑝) ∈ (𝐹𝐴))))
41 opelxp 5391 . . . . . 6 (⟨(1st𝑝), (2nd𝑝)⟩ ∈ ((𝐹𝐴) × (𝐹𝐴)) ↔ ((1st𝑝) ∈ (𝐹𝐴) ∧ (2nd𝑝) ∈ (𝐹𝐴)))
4240, 41syl6bbr 281 . . . . 5 ((𝐹 Fn 𝑋𝐴𝑋) → (⟨(1st𝑝), (2nd𝑝)⟩ ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ ⟨(1st𝑝), (2nd𝑝)⟩ ∈ ((𝐹𝐴) × (𝐹𝐴))))
4342adantr 474 . . . 4 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ (V × V)) → (⟨(1st𝑝), (2nd𝑝)⟩ ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ ⟨(1st𝑝), (2nd𝑝)⟩ ∈ ((𝐹𝐴) × (𝐹𝐴))))
44 1st2nd2 7484 . . . . . 6 (𝑝 ∈ (V × V) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
4544adantl 475 . . . . 5 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ (V × V)) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
4645eleq1d 2844 . . . 4 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ ⟨(1st𝑝), (2nd𝑝)⟩ ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩)))
4745eleq1d 2844 . . . 4 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ((𝐹𝐴) × (𝐹𝐴)) ↔ ⟨(1st𝑝), (2nd𝑝)⟩ ∈ ((𝐹𝐴) × (𝐹𝐴))))
4843, 46, 473bitr4d 303 . . 3 (((𝐹 Fn 𝑋𝐴𝑋) ∧ 𝑝 ∈ (V × V)) → (𝑝 ∈ ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) ↔ 𝑝 ∈ ((𝐹𝐴) × (𝐹𝐴))))
4922, 25, 48eqrdav 2777 . 2 ((𝐹 Fn 𝑋𝐴𝑋) → ran (𝑥𝐴, 𝑦𝐴 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) = ((𝐹𝐴) × (𝐹𝐴)))
506, 49eqtrd 2814 1 ((𝐹 Fn 𝑋𝐴𝑋) → ((𝑥𝑋, 𝑦𝑋 ↦ ⟨(𝐹𝑥), (𝐹𝑦)⟩) “ (𝐴 × 𝐴)) = ((𝐹𝐴) × (𝐹𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386   = wceq 1601  wtru 1602  wcel 2107  wrex 3091  Vcvv 3398  wss 3792  cop 4404  cmpt 4965   × cxp 5353  ran crn 5356  cres 5357  cima 5358   Fn wfn 6130  cfv 6135  cmpt2 6924  1st c1st 7443  2nd c2nd 7444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-oprab 6926  df-mpt2 6927  df-1st 7445  df-2nd 7446
This theorem is referenced by:  fmucnd  22504
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