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Mirrors > Home > MPE Home > Th. List > ovmpodv2 | Structured version Visualization version GIF version |
Description: Alternate deduction version of ovmpo 7303, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.) |
Ref | Expression |
---|---|
ovmpodv2.1 | ⊢ (𝜑 → 𝐴 ∈ 𝐶) |
ovmpodv2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) |
ovmpodv2.3 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) |
ovmpodv2.4 | ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) |
Ref | Expression |
---|---|
ovmpodv2 | ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2821 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)) | |
2 | ovmpodv2.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐶) | |
3 | ovmpodv2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐵 ∈ 𝐷) | |
4 | ovmpodv2.3 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 ∈ 𝑉) | |
5 | ovmpodv2.4 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝑅 = 𝑆) | |
6 | 5 | eqeq2d 2831 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅 ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
7 | 6 | biimpd 231 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → ((𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑅 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
8 | nfmpo1 7227 | . . . 4 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
9 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
10 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
11 | 9, 8, 10 | nfov 7179 | . . . . 5 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) |
12 | 11 | nfeq1 2992 | . . . 4 ⊢ Ⅎ𝑥(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆 |
13 | nfmpo2 7228 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) | |
14 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑦𝐴 | |
15 | nfcv 2976 | . . . . . 6 ⊢ Ⅎ𝑦𝐵 | |
16 | 14, 13, 15 | nfov 7179 | . . . . 5 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) |
17 | 16 | nfeq1 2992 | . . . 4 ⊢ Ⅎ𝑦(𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆 |
18 | 2, 3, 4, 7, 8, 12, 13, 17 | ovmpodf 7299 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
19 | 1, 18 | mpd 15 | . 2 ⊢ (𝜑 → (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆) |
20 | oveq 7155 | . . 3 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵)) | |
21 | 20 | eqeq1d 2822 | . 2 ⊢ (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → ((𝐴𝐹𝐵) = 𝑆 ↔ (𝐴(𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅)𝐵) = 𝑆)) |
22 | 19, 21 | syl5ibrcom 249 | 1 ⊢ (𝜑 → (𝐹 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐷 ↦ 𝑅) → (𝐴𝐹𝐵) = 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 (class class class)co 7149 ∈ cmpo 7151 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 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 7152 df-oprab 7153 df-mpo 7154 |
This theorem is referenced by: coaval 17321 xpcco 17426 marrepval 21164 marrepeval 21165 marepveval 21170 submaval 21183 submaeval 21184 minmar1val 21250 minmar1eval 21251 nbgrval 27114 |
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