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Theorem dfoprab4f 7743
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 1906 . . . . 5 𝑥 𝑤 = ⟨𝑡, 𝑢
2 dfoprab4f.x . . . . . 6 𝑥𝜑
3 nfs1v 2264 . . . . . 6 𝑥[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
42, 3nfbi 1895 . . . . 5 𝑥(𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
51, 4nfim 1888 . . . 4 𝑥(𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
6 opeq1 4795 . . . . . 6 (𝑥 = 𝑡 → ⟨𝑥, 𝑢⟩ = ⟨𝑡, 𝑢⟩)
76eqeq2d 2829 . . . . 5 (𝑥 = 𝑡 → (𝑤 = ⟨𝑥, 𝑢⟩ ↔ 𝑤 = ⟨𝑡, 𝑢⟩))
8 sbequ12 2243 . . . . . 6 (𝑥 = 𝑡 → ([𝑢 / 𝑦]𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
98bibi2d 344 . . . . 5 (𝑥 = 𝑡 → ((𝜑 ↔ [𝑢 / 𝑦]𝜓) ↔ (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
107, 9imbi12d 346 . . . 4 (𝑥 = 𝑡 → ((𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓)) ↔ (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))))
11 nfv 1906 . . . . . 6 𝑦 𝑤 = ⟨𝑥, 𝑢
12 dfoprab4f.y . . . . . . 7 𝑦𝜑
13 nfs1v 2264 . . . . . . 7 𝑦[𝑢 / 𝑦]𝜓
1412, 13nfbi 1895 . . . . . 6 𝑦(𝜑 ↔ [𝑢 / 𝑦]𝜓)
1511, 14nfim 1888 . . . . 5 𝑦(𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
16 opeq2 4796 . . . . . . 7 (𝑦 = 𝑢 → ⟨𝑥, 𝑦⟩ = ⟨𝑥, 𝑢⟩)
1716eqeq2d 2829 . . . . . 6 (𝑦 = 𝑢 → (𝑤 = ⟨𝑥, 𝑦⟩ ↔ 𝑤 = ⟨𝑥, 𝑢⟩))
18 sbequ12 2243 . . . . . . 7 (𝑦 = 𝑢 → (𝜓 ↔ [𝑢 / 𝑦]𝜓))
1918bibi2d 344 . . . . . 6 (𝑦 = 𝑢 → ((𝜑𝜓) ↔ (𝜑 ↔ [𝑢 / 𝑦]𝜓)))
2017, 19imbi12d 346 . . . . 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 2232 . . . 4 (𝑤 = ⟨𝑥, 𝑢⟩ → (𝜑 ↔ [𝑢 / 𝑦]𝜓))
235, 10, 22chvarfv 2232 . . 3 (𝑤 = ⟨𝑡, 𝑢⟩ → (𝜑 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
2423dfoprab4 7742 . 2 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
25 nfv 1906 . . 3 𝑡((𝑥𝐴𝑦𝐵) ∧ 𝜓)
26 nfv 1906 . . 3 𝑢((𝑥𝐴𝑦𝐵) ∧ 𝜓)
27 nfv 1906 . . . 4 𝑥(𝑡𝐴𝑢𝐵)
2827, 3nfan 1891 . . 3 𝑥((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
29 nfv 1906 . . . 4 𝑦(𝑡𝐴𝑢𝐵)
3013nfsbv 2340 . . . 4 𝑦[𝑡 / 𝑥][𝑢 / 𝑦]𝜓
3129, 30nfan 1891 . . 3 𝑦((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)
32 eleq1w 2892 . . . . 5 (𝑥 = 𝑡 → (𝑥𝐴𝑡𝐴))
33 eleq1w 2892 . . . . 5 (𝑦 = 𝑢 → (𝑦𝐵𝑢𝐵))
3432, 33bi2anan9 635 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → ((𝑥𝐴𝑦𝐵) ↔ (𝑡𝐴𝑢𝐵)))
3518, 8sylan9bbr 511 . . . 4 ((𝑥 = 𝑡𝑦 = 𝑢) → (𝜓 ↔ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓))
3634, 35anbi12d 630 . . 3 ((𝑥 = 𝑡𝑦 = 𝑢) → (((𝑥𝐴𝑦𝐵) ∧ 𝜓) ↔ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)))
3725, 26, 28, 31, 36cbvoprab12 7232 . 2 {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)} = {⟨⟨𝑡, 𝑢⟩, 𝑧⟩ ∣ ((𝑡𝐴𝑢𝐵) ∧ [𝑡 / 𝑥][𝑢 / 𝑦]𝜓)}
3824, 37eqtr4i 2844 1 {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (𝐴 × 𝐵) ∧ 𝜑)} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝜓)}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wnf 1775  [wsb 2060  wcel 2105  cop 4563  {copab 5119   × cxp 5546  {coprab 7146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fv 6356  df-oprab 7149  df-1st 7678  df-2nd 7679
This theorem is referenced by: (None)
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