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Theorem dfoprab4f 7738
 Description: Operation class abstraction expressed without existential quantifiers. (Contributed by NM, 20-Dec-2008.) Remove unnecessary distinct variable conditions. (Revised by David Abernethy, 19-Jun-2012.) (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 1915 . . . . 5 𝑥 𝑤 = ⟨𝑡, 𝑢
2 dfoprab4f.x . . . . . 6 𝑥𝜑
3 nfs1v 2157 . . . . . 6 𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
42, 3nfbi 1904 . . . . 5 𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
51, 4nfim 1897 . . . 4 𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6 opeq1 4763 . . . . . 6 (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
76eqeq2d 2809 . . . . 5 (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8 sbequ12 2250 . . . . . 6 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
98bibi2d 346 . . . . 5 (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
107, 9imbi12d 348 . . . 4 (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11 nfv 1915 . . . . . 6 𝑦 𝑤 = ⟨𝑥, 𝑢
12 dfoprab4f.y . . . . . . 7 𝑦𝜑
13 nfs1v 2157 . . . . . . 7 𝑦[𝑢 / 𝑦]𝜓
1412, 13nfbi 1904 . . . . . 6 𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
1511, 14nfim 1897 . . . . 5 𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16 opeq2 4765 . . . . . . 7 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
1716eqeq2d 2809 . . . . . 6 (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18 sbequ12 2250 . . . . . . 7 (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
1918bibi2d 346 . . . . . 6 (𝑦 = 𝑢 → ((𝜑𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
2017, 19imbi12d 348 . . . . 5 (𝑦 = 𝑢 → ((𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓)) ↔ (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))))
21 dfoprab4f.1 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (𝜑𝜓))
2215, 20, 21chvarfv 2240 . . . 4 (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
235, 10, 22chvarfv 2240 . . 3 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
2423dfoprab4 7737 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25 nfv 1915 . . 3 𝑡((𝑥𝐴𝑦𝐵) ∧ 𝜓)
26 nfv 1915 . . 3 𝑢((𝑥𝐴𝑦𝐵) ∧ 𝜓)
27 nfv 1915 . . . 4 𝑥(𝑡𝐴𝑢𝐵)
2827, 3nfan 1900 . . 3 𝑥((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29 nfv 1915 . . . 4 𝑦(𝑡𝐴𝑢𝐵)
3013nfsbv 2338 . . . 4 𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
3129, 30nfan 1900 . . 3 𝑦((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32 eleq1w 2872 . . . . 5 (𝑥 = 𝑡 → (𝑥𝐴𝑡𝐴))
33 eleq1w 2872 . . . . 5 (𝑦 = 𝑢 → (𝑦𝐵𝑢𝐵))
3432, 33bi2anan9 638 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → ((𝑥𝐴𝑦𝐵) ↔ (𝑡𝐴𝑢𝐵)))
3518, 8sylan9bbr 514 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
3634, 35anbi12d 633 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
3725, 26, 28, 31, 36cbvoprab12 7222 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
3824, 37eqtr4i 2824 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538  Ⅎwnf 1785  [wsb 2069   ∈ wcel 2111  ⟨cop 4531  {copab 5092   × cxp 5517  {coprab 7136 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 7443 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-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-iota 6283  df-fun 6326  df-fv 6332  df-oprab 7139  df-1st 7673  df-2nd 7674 This theorem is referenced by: (None)
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