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Theorem elrnmpt2res 6816
Description: Membership in the range of a restricted operation class abstraction. (Contributed by Thierry Arnoux, 25-May-2019.)
Hypothesis
Ref Expression
rngop.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
elrnmpt2res (𝐷𝑉 → (𝐷 ∈ ran (𝐹𝑅) ↔ ∃𝑥𝐴𝑦𝐵 (𝐷 = 𝐶𝑥𝑅𝑦)))
Distinct variable groups:   𝑦,𝐴   𝑥,𝑦,𝐷   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem elrnmpt2res
Dummy variables 𝑧 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2655 . . . . . 6 (𝑧 = 𝐷 → (𝑧 = 𝐶𝐷 = 𝐶))
21anbi1d 741 . . . . 5 (𝑧 = 𝐷 → ((𝑧 = 𝐶𝑥𝑅𝑦) ↔ (𝐷 = 𝐶𝑥𝑅𝑦)))
32anbi2d 740 . . . 4 (𝑧 = 𝐷 → (((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝐷 = 𝐶𝑥𝑅𝑦))))
432exbidv 1892 . . 3 (𝑧 = 𝐷 → (∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦)) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐷 = 𝐶𝑥𝑅𝑦))))
5 an12 855 . . . . . . . . . 10 ((𝑝𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑝𝑅 ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))))
6 an12 855 . . . . . . . . . . . 12 ((𝑝𝑅 ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑝𝑅𝑧 = 𝐶)))
7 ancom 465 . . . . . . . . . . . . . 14 ((𝑧 = 𝐶𝑝𝑅) ↔ (𝑝𝑅𝑧 = 𝐶))
8 eleq1 2718 . . . . . . . . . . . . . . . 16 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
9 df-br 4686 . . . . . . . . . . . . . . . 16 (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
108, 9syl6bbr 278 . . . . . . . . . . . . . . 15 (𝑝 = ⟨𝑥, 𝑦⟩ → (𝑝𝑅𝑥𝑅𝑦))
1110anbi2d 740 . . . . . . . . . . . . . 14 (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑧 = 𝐶𝑝𝑅) ↔ (𝑧 = 𝐶𝑥𝑅𝑦)))
127, 11syl5bbr 274 . . . . . . . . . . . . 13 (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝𝑅𝑧 = 𝐶) ↔ (𝑧 = 𝐶𝑥𝑅𝑦)))
1312anbi2d 740 . . . . . . . . . . . 12 (𝑝 = ⟨𝑥, 𝑦⟩ → (((𝑥𝐴𝑦𝐵) ∧ (𝑝𝑅𝑧 = 𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))))
146, 13syl5bb 272 . . . . . . . . . . 11 (𝑝 = ⟨𝑥, 𝑦⟩ → ((𝑝𝑅 ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))))
1514pm5.32i 670 . . . . . . . . . 10 ((𝑝 = ⟨𝑥, 𝑦⟩ ∧ (𝑝𝑅 ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))))
165, 15bitri 264 . . . . . . . . 9 ((𝑝𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))))
17162exbii 1815 . . . . . . . 8 (∃𝑥𝑦(𝑝𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))) ↔ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))))
18 19.42vv 1923 . . . . . . . 8 (∃𝑥𝑦(𝑝𝑅 ∧ (𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))) ↔ (𝑝𝑅 ∧ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))))
1917, 18bitr3i 266 . . . . . . 7 (∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))) ↔ (𝑝𝑅 ∧ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))))
2019opabbii 4750 . . . . . 6 {⟨𝑝, 𝑧⟩ ∣ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦)))} = {⟨𝑝, 𝑧⟩ ∣ (𝑝𝑅 ∧ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)))}
21 dfoprab2 6743 . . . . . 6 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))} = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦)))}
22 rngop.1 . . . . . . . . 9 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
23 df-mpt2 6695 . . . . . . . . 9 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
24 dfoprab2 6743 . . . . . . . . 9 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)} = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))}
2522, 23, 243eqtri 2677 . . . . . . . 8 𝐹 = {⟨𝑝, 𝑧⟩ ∣ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))}
2625reseq1i 5424 . . . . . . 7 (𝐹𝑅) = ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅)
27 resopab 5481 . . . . . . 7 ({⟨𝑝, 𝑧⟩ ∣ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶))} ↾ 𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝𝑅 ∧ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)))}
2826, 27eqtri 2673 . . . . . 6 (𝐹𝑅) = {⟨𝑝, 𝑧⟩ ∣ (𝑝𝑅 ∧ ∃𝑥𝑦(𝑝 = ⟨𝑥, 𝑦⟩ ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)))}
2920, 21, 283eqtr4ri 2684 . . . . 5 (𝐹𝑅) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))}
3029rneqi 5384 . . . 4 ran (𝐹𝑅) = ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))}
31 rnoprab 6785 . . . 4 ran {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))} = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))}
3230, 31eqtri 2673 . . 3 ran (𝐹𝑅) = {𝑧 ∣ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝑧 = 𝐶𝑥𝑅𝑦))}
334, 32elab2g 3385 . 2 (𝐷𝑉 → (𝐷 ∈ ran (𝐹𝑅) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐷 = 𝐶𝑥𝑅𝑦))))
34 r2ex 3090 . 2 (∃𝑥𝐴𝑦𝐵 (𝐷 = 𝐶𝑥𝑅𝑦) ↔ ∃𝑥𝑦((𝑥𝐴𝑦𝐵) ∧ (𝐷 = 𝐶𝑥𝑅𝑦)))
3533, 34syl6bbr 278 1 (𝐷𝑉 → (𝐷 ∈ ran (𝐹𝑅) ↔ ∃𝑥𝐴𝑦𝐵 (𝐷 = 𝐶𝑥𝑅𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wrex 2942  cop 4216   class class class wbr 4685  {copab 4745  ran crn 5144  cres 5145  {coprab 6691  cmpt2 6692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-oprab 6694  df-mpt2 6695
This theorem is referenced by: (None)
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