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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvmlift2lem4 | Structured version Visualization version GIF version |
Description: Lemma for cvmlift2 35301. (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 7438 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑧) = (𝑋𝐺𝑧)) | |
2 | 1 | mpteq2dv 5250 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧))) |
3 | 2 | eqeq2d 2746 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑥𝐺𝑧)) ↔ (𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)))) |
4 | fveq2 6907 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐻‘𝑥) = (𝐻‘𝑋)) | |
5 | 4 | eqeq2d 2746 | . . . . 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 7390 | . . 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 6909 | . 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 6907 | . 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 6920 | . 2 ⊢ ((℩𝑓 ∈ (II Cn 𝐶)((𝐹 ∘ 𝑓) = (𝑧 ∈ (0[,]1) ↦ (𝑋𝐺𝑧)) ∧ (𝑓‘0) = (𝐻‘𝑋)))‘𝑌) ∈ V | |
12 | 8, 9, 10, 11 | ovmpo 7593 | 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 395 = wceq 1537 ∈ wcel 2106 ∪ cuni 4912 ↦ cmpt 5231 ∘ ccom 5693 ‘cfv 6563 ℩crio 7387 (class class class)co 7431 ∈ cmpo 7433 0cc0 11153 1c1 11154 [,]cicc 13387 Cn ccn 23248 ×t ctx 23584 IIcii 24915 CovMap ccvm 35240 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 |
This theorem is referenced by: cvmlift2lem6 35293 cvmlift2lem8 35295 |
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