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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmlift2lem4 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift2 32563. (Contributed by Mario Carneiro, 1-Jun-2015.) |
Ref | Expression |
---|---|
cvmlift2.b | ⊢ 𝐵 = ∪ 𝐶 |
cvmlift2.f | ⊢ (𝜑 → 𝐹 ∈ (𝐶 CovMap 𝐽)) |
cvmlift2.g | ⊢ (𝜑 → 𝐺 ∈ ((II ×t II) Cn 𝐽)) |
cvmlift2.p | ⊢ (𝜑 → 𝑃 ∈ 𝐵) |
cvmlift2.i | ⊢ (𝜑 → (𝐹‘𝑃) = (0𝐺0)) |
cvmlift2.h | ⊢ 𝐻 = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑧𝐺0)) ∧ (𝑓‘0) = 𝑃)) |
cvmlift2.k | ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) |
Ref | Expression |
---|---|
cvmlift2lem4 | ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋𝐾𝑌) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7162 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑧) = (𝑋𝐺𝑧)) | |
2 | 1 | mpteq2dv 5161 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))) |
3 | 2 | eqeq2d 2832 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ↔ (𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)))) |
4 | fveq2 6669 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋)) | |
5 | 4 | eqeq2d 2832 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑓‘0) = (𝐻‘𝑥) ↔ (𝑓‘0) = (𝐻‘𝑋))) |
6 | 3, 5 | anbi12d 632 | . . . 4 ⊢ (𝑥 = 𝑋 → (((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)) ↔ ((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))) |
7 | 6 | riotabidv 7115 | . . 3 ⊢ (𝑥 = 𝑋 → (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥))) = (℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))) |
8 | 7 | fveq1d 6671 | . 2 ⊢ (𝑥 = 𝑋 → ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑦)) |
9 | fveq2 6669 | . 2 ⊢ (𝑦 = 𝑌 → ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑦) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌)) | |
10 | cvmlift2.k | . 2 ⊢ 𝐾 = (𝑥 ∈ (0[,]1), 𝑦 ∈ (0[,]1) ↦ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑥)))‘𝑦)) | |
11 | fvex 6682 | . 2 ⊢ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌) ∈ V | |
12 | 8, 9, 10, 11 | ovmpo 7309 | 1 ⊢ ((𝑋 ∈ (0[,]1) ∧ 𝑌 ∈ (0[,]1)) → (𝑋𝐾𝑌) = ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∪ cuni 4837 ↦ cmpt 5145 ∘ ccom 5558 ‘cfv 6354 ℩crio 7112 (class class class)co 7155 ∈ cmpo 7157 0cc0 10536 1c1 10537 [,]cicc 12740 Cn ccn 21831 ×t ctx 22167 IIcii 23482 CovMap ccvm 32502 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-iota 6313 df-fun 6356 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 |
This theorem is referenced by: cvmlift2lem6 32555 cvmlift2lem8 32557 |
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