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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 6881 | . . . . . 6 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑧) = (𝐹‘𝑧)) | |
4 | fveq1 6881 | . . . . . 6 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑧) = (𝐺‘𝑧)) | |
5 | eleq12 2815 | . . . . . 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 6881 | . . . . . . . 8 ⊢ (𝑥 = 𝐹 → (𝑥‘𝑤) = (𝐹‘𝑤)) | |
8 | fveq1 6881 | . . . . . . . 8 ⊢ (𝑦 = 𝐺 → (𝑦‘𝑤) = (𝐺‘𝑤)) | |
9 | 7, 8 | eqeqan12d 2738 | . . . . . . 7 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑥‘𝑤) = (𝑦‘𝑤) ↔ (𝐹‘𝑤) = (𝐺‘𝑤))) |
10 | 9 | imbi2d 340 | . . . . . 6 ⊢ ((𝑥 = 𝐹 ∧ 𝑦 = 𝐺) → ((𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)) ↔ (𝑧 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
11 | 10 | ralbidv 3169 | . . . . 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 3170 | . . 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 5525 | . 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 583 | 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 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∀wral 3053 ∃wrex 3062 class class class wbr 5139 {copab 5201 dom cdm 5667 Oncon0 6355 ‘cfv 6534 (class class class)co 7402 CNF ccnf 9653 |
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 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-iota 6486 df-fv 6542 |
This theorem is referenced by: oemapvali 9676 |
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