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Theorem ovmpt2df 6834
 Description: Alternate deduction version of ovmpt2 6838, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
ovmpt2df.1 (𝜑𝐴𝐶)
ovmpt2df.2 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
ovmpt2df.3 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
ovmpt2df.4 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
ovmpt2df.5 𝑥𝐹
ovmpt2df.6 𝑥𝜓
ovmpt2df.7 𝑦𝐹
ovmpt2df.8 𝑦𝜓
Assertion
Ref Expression
ovmpt2df (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 ovmpt2df
StepHypRef Expression
1 nfv 1883 . 2 𝑥𝜑
2 ovmpt2df.5 . . . 4 𝑥𝐹
3 nfmpt21 6764 . . . 4 𝑥(𝑥𝐶, 𝑦𝐷𝑅)
42, 3nfeq 2805 . . 3 𝑥 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
5 ovmpt2df.6 . . 3 𝑥𝜓
64, 5nfim 1865 . 2 𝑥(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
7 ovmpt2df.1 . . . 4 (𝜑𝐴𝐶)
8 elex 3243 . . . 4 (𝐴𝐶𝐴 ∈ V)
97, 8syl 17 . . 3 (𝜑𝐴 ∈ V)
10 isset 3238 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
119, 10sylib 208 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
12 ovmpt2df.2 . . . . 5 ((𝜑𝑥 = 𝐴) → 𝐵𝐷)
13 elex 3243 . . . . 5 (𝐵𝐷𝐵 ∈ V)
1412, 13syl 17 . . . 4 ((𝜑𝑥 = 𝐴) → 𝐵 ∈ V)
15 isset 3238 . . . 4 (𝐵 ∈ V ↔ ∃𝑦 𝑦 = 𝐵)
1614, 15sylib 208 . . 3 ((𝜑𝑥 = 𝐴) → ∃𝑦 𝑦 = 𝐵)
17 nfv 1883 . . . 4 𝑦(𝜑𝑥 = 𝐴)
18 ovmpt2df.7 . . . . . 6 𝑦𝐹
19 nfmpt22 6765 . . . . . 6 𝑦(𝑥𝐶, 𝑦𝐷𝑅)
2018, 19nfeq 2805 . . . . 5 𝑦 𝐹 = (𝑥𝐶, 𝑦𝐷𝑅)
21 ovmpt2df.8 . . . . 5 𝑦𝜓
2220, 21nfim 1865 . . . 4 𝑦(𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)
23 oveq 6696 . . . . . 6 (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
24 simprl 809 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥 = 𝐴)
25 simprr 811 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦 = 𝐵)
2624, 25oveq12d 6708 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵))
277adantr 480 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐴𝐶)
2824, 27eqeltrd 2730 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑥𝐶)
2912adantrr 753 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝐵𝐷)
3025, 29eqeltrd 2730 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑦𝐷)
31 ovmpt2df.3 . . . . . . . . . 10 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → 𝑅𝑉)
32 eqid 2651 . . . . . . . . . . 11 (𝑥𝐶, 𝑦𝐷𝑅) = (𝑥𝐶, 𝑦𝐷𝑅)
3332ovmpt4g 6825 . . . . . . . . . 10 ((𝑥𝐶𝑦𝐷𝑅𝑉) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3428, 30, 31, 33syl3anc 1366 . . . . . . . . 9 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝑥(𝑥𝐶, 𝑦𝐷𝑅)𝑦) = 𝑅)
3526, 34eqtr3d 2687 . . . . . . . 8 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) = 𝑅)
3635eqeq2d 2661 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) ↔ (𝐴𝐹𝐵) = 𝑅))
37 ovmpt2df.4 . . . . . . 7 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = 𝑅𝜓))
3836, 37sylbid 230 . . . . . 6 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → ((𝐴𝐹𝐵) = (𝐴(𝑥𝐶, 𝑦𝐷𝑅)𝐵) → 𝜓))
3923, 38syl5 34 . . . . 5 ((𝜑 ∧ (𝑥 = 𝐴𝑦 = 𝐵)) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
4039expr 642 . . . 4 ((𝜑𝑥 = 𝐴) → (𝑦 = 𝐵 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)))
4117, 22, 40exlimd 2125 . . 3 ((𝜑𝑥 = 𝐴) → (∃𝑦 𝑦 = 𝐵 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓)))
4216, 41mpd 15 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
431, 6, 11, 42exlimdd 2126 1 (𝜑 → (𝐹 = (𝑥𝐶, 𝑦𝐷𝑅) → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1523  ∃wex 1744  Ⅎwnf 1748   ∈ wcel 2030  Ⅎwnfc 2780  Vcvv 3231  (class class class)co 6690   ↦ 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-sbc 3469  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-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695 This theorem is referenced by:  ovmpt2dv  6835  ovmpt2dv2  6836
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