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Theorem dfoprab4f 7178
Description: Operation class abstraction expressed without existential quantifiers. (Unnecessary distinct variable restrictions were removed by David Abernethy, 19-Jun-2012.) (Contributed by NM, 20-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypotheses
Ref Expression
dfoprab4f.x 𝑥𝜑
dfoprab4f.y 𝑦𝜑
dfoprab4f.1 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
Assertion
Ref Expression
dfoprab4f {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
Distinct variable groups:   𝑥,𝑤,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦   𝑤,𝐵,𝑥,𝑦   𝜓,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝜓(𝑥,𝑦,𝑧)   𝐴(𝑧)   𝐵(𝑧)

Proof of Theorem dfoprab4f
Dummy variables 𝑢 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1840 . . . . 5 𝑥 𝑤 = ⟨𝑡, 𝑢
2 dfoprab4f.x . . . . . 6 𝑥𝜑
3 nfs1v 2436 . . . . . 6 𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
42, 3nfbi 1830 . . . . 5 𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
51, 4nfim 1822 . . . 4 𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6 opeq1 4375 . . . . . 6 (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
76eqeq2d 2631 . . . . 5 (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8 sbequ12 2108 . . . . . 6 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
98bibi2d 332 . . . . 5 (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
107, 9imbi12d 334 . . . 4 (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11 nfv 1840 . . . . . 6 𝑦 𝑤 = ⟨𝑥, 𝑢
12 dfoprab4f.y . . . . . . 7 𝑦𝜑
13 nfs1v 2436 . . . . . . 7 𝑦[𝑢 / 𝑦]𝜓
1412, 13nfbi 1830 . . . . . 6 𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
1511, 14nfim 1822 . . . . 5 𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16 opeq2 4376 . . . . . . 7 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
1716eqeq2d 2631 . . . . . 6 (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18 sbequ12 2108 . . . . . . 7 (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
1918bibi2d 332 . . . . . 6 (𝑦 = 𝑢 → ((𝜑𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
2017, 19imbi12d 334 . . . . 5 (𝑦 = 𝑢 → ((𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓)) ↔ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))))
21 dfoprab4f.1 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
2215, 20, 21chvar 2261 . . . 4 (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
235, 10, 22chvar 2261 . . 3 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
2423dfoprab4 7177 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25 nfv 1840 . . 3 𝑡((𝑥𝐴𝑦𝐵) ∧ 𝜓)
26 nfv 1840 . . 3 𝑢((𝑥𝐴𝑦𝐵) ∧ 𝜓)
27 nfv 1840 . . . 4 𝑥(𝑡𝐴𝑢𝐵)
2827, 3nfan 1825 . . 3 𝑥((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29 nfv 1840 . . . 4 𝑦(𝑡𝐴𝑢𝐵)
3013nfsb 2439 . . . 4 𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
3129, 30nfan 1825 . . 3 𝑦((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32 eleq1 2686 . . . . 5 (𝑥 = 𝑡 → (𝑥𝐴𝑡𝐴))
33 eleq1 2686 . . . . 5 (𝑦 = 𝑢 → (𝑦𝐵𝑢𝐵))
3432, 33bi2anan9 916 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → ((𝑥𝐴𝑦𝐵) ↔ (𝑡𝐴𝑢𝐵)))
3518, 8sylan9bbr 736 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
3634, 35anbi12d 746 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
3725, 26, 28, 31, 36cbvoprab12 6689 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
3824, 37eqtr4i 2646 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wnf 1705  [wsb 1877  wcel 1987  cop 4159  {copab 4677   × cxp 5077  {coprab 6611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5815  df-fun 5854  df-fv 5860  df-oprab 6614  df-1st 7120  df-2nd 7121
This theorem is referenced by: (None)
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