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Theorem ovmpodf 7298
Description: Alternate deduction version of ovmpo 7302, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpodf.1 (𝜑𝐴𝐶)
ovmpodf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpodf.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpodf.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
ovmpodf.5 𝑥𝐹
ovmpodf.6 𝑥𝜓
ovmpodf.7 𝑦𝐹
ovmpodf.8 𝑦𝜓
Assertion
Ref Expression
ovmpodf (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜑,𝑥,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝐹(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem ovmpodf
StepHypRef Expression
1 nfv 1908 . 2 𝑥𝜑
2 ovmpodf.5 . . . 4 𝑥𝐹
3 nfmpo1 7226 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
42, 3nfeq 2989 . . 3 𝑥 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
5 ovmpodf.6 . . 3 𝑥𝜓
64, 5nfim 1890 . 2 𝑥(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
7 ovmpodf.1 . . . 4 (𝜑𝐴𝐶)
87elexd 3513 . . 3 (𝜑𝐴 ∈ V)
9 isset 3505 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 220 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 nfv 1908 . . 3 𝑦(𝜑𝑥 = 𝐴)
12 ovmpodf.7 . . . . 5 𝑦𝐹
13 nfmpo2 7227 . . . . 5 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
1412, 13nfeq 2989 . . . 4 𝑦 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
15 ovmpodf.8 . . . 4 𝑦𝜓
1614, 15nfim 1890 . . 3 𝑦(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
17 ovmpodf.2 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
1817elexd 3513 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
19 isset 3505 . . . 4 (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
2018, 19sylib 220 . . 3 ((𝜑𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
21 oveq 7154 . . . . 5 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
22 simprl 769 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥 = 𝐴)
23 simprr 771 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦 = 𝐵)
2422, 23oveq12d 7166 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
257adantr 483 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐴𝐶)
2622, 25eqeltrd 2911 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥𝐶)
2717adantrr 715 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵𝐷)
2823, 27eqeltrd 2911 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦𝐷)
29 ovmpodf.3 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
30 eqid 2819 . . . . . . . . . 10 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
3130ovmpt4g 7289 . . . . . . . . 9 ((𝑥𝐶𝑦𝐷𝑅𝑉) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3226, 28, 29, 31syl3anc 1365 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3324, 32eqtr3d 2856 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅)
3433eqeq2d 2830 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
35 ovmpodf.4 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
3634, 35sylbid 242 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) → 𝜓))
3721, 36syl5 34 . . . 4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
3837expr 459 . . 3 ((𝜑𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)))
3911, 16, 20, 38exlimimdd 2211 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
401, 6, 10, 39exlimdd 2212 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1530  wex 1773  wnf 1777  wcel 2107  wnfc 2959  Vcvv 3493  (class class class)co 7148  cmpo 7150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153
This theorem is referenced by:  ovmpodv  7299  ovmpodv2  7300
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