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Mirrors > Home > MPE Home > Th. List > oemapval | Structured version Visualization version GIF version |
Description: Value of the relation 𝑇. (Contributed by Mario Carneiro, 28-May-2015.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
oemapval.t | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
oemapval.f | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
oemapval.g | ⊢ (𝜑 → 𝐺 ∈ 𝑆) |
Ref | Expression |
---|---|
oemapval | ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oemapval.f | . 2 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
2 | oemapval.g | . 2 ⊢ (𝜑 → 𝐺 ∈ 𝑆) | |
3 | fveq1 6662 | . . . . . 6 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑧) = (𝐹‘𝑧)) | |
4 | fveq1 6662 | . . . . . 6 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑧) = (𝐺‘𝑧)) | |
5 | eleq12 2899 | . . . . . 6 ⊢ (((𝑥‘𝑧) = (𝐹‘𝑧) ∧ (𝑦‘𝑧) = (𝐺‘𝑧)) → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧))) | |
6 | 3, 4, 5 | syl2an 595 | . . . . 5 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ↔ (𝐹‘𝑧) ∈ (𝐺‘𝑧))) |
7 | fveq1 6662 | . . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤)) | |
8 | fveq1 6662 | . . . . . . . 8 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑤) = (𝐺‘𝑤)) | |
9 | 7, 8 | eqeqan12d 2835 | . . . . . . 7 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝐹‘𝑤) = (𝐺‘𝑤))) |
10 | 9 | imbi2d 342 | . . . . . 6 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
11 | 10 | ralbidv 3194 | . . . . 5 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
12 | 6, 11 | anbi12d 630 | . . . 4 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) |
13 | 12 | rexbidv 3294 | . . 3 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → (∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤))) ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) |
14 | oemapval.t | . . 3 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} | |
15 | 13, 14 | brabga 5412 | . 2 ⊢ ((𝐹 ∈ 𝑆 ∧ 𝐺 ∈ 𝑆) → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) |
16 | 1, 2, 15 | syl2anc 584 | 1 ⊢ (𝜑 → (𝐹𝑇𝐺 ↔ ∃𝑧 ∈ 𝐵 ((𝐹‘𝑧) ∈ (𝐺‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 class class class wbr 5057 {copab 5119 dom cdm 5548 Oncon0 6184 ‘cfv 6348 (class class class)co 7145 CNF ccnf 9112 |
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-pr 5320 |
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-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-iota 6307 df-fv 6356 |
This theorem is referenced by: oemapvali 9135 |
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