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Mirrors > Home > MPE Home > Th. List > ishtpyd | Structured version Visualization version GIF version |
Description: Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.) |
Ref | Expression |
---|---|
ishtpy.1 | ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
ishtpy.3 | ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) |
ishtpy.4 | ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) |
ishtpyd.1 | ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾)) |
ishtpyd.2 | ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻0) = (𝐹‘𝑠)) |
ishtpyd.3 | ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻1) = (𝐺‘𝑠)) |
Ref | Expression |
---|---|
ishtpyd | ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ishtpyd.1 | . 2 ⊢ (𝜑 → 𝐻 ∈ ((𝐽 ×t II) Cn 𝐾)) | |
2 | ishtpyd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻0) = (𝐹‘𝑠)) | |
3 | ishtpyd.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → (𝑠𝐻1) = (𝐺‘𝑠)) | |
4 | 2, 3 | jca 512 | . . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑋) → ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))) |
5 | 4 | ralrimiva 3143 | . 2 ⊢ (𝜑 → ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))) |
6 | ishtpy.1 | . . 3 ⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) | |
7 | ishtpy.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | |
8 | ishtpy.4 | . . 3 ⊢ (𝜑 → 𝐺 ∈ (𝐽 Cn 𝐾)) | |
9 | 6, 7, 8 | ishtpy 24335 | . 2 ⊢ (𝜑 → (𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺) ↔ (𝐻 ∈ ((𝐽 ×t II) Cn 𝐾) ∧ ∀𝑠 ∈ 𝑋 ((𝑠𝐻0) = (𝐹‘𝑠) ∧ (𝑠𝐻1) = (𝐺‘𝑠))))) |
10 | 1, 5, 9 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝐻 ∈ (𝐹(𝐽 Htpy 𝐾)𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3064 ‘cfv 6496 (class class class)co 7357 0cc0 11051 1c1 11052 TopOnctopon 22259 Cn ccn 22575 ×t ctx 22911 IIcii 24238 Htpy chtpy 24330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-map 8767 df-top 22243 df-topon 22260 df-cn 22578 df-htpy 24333 |
This theorem is referenced by: htpycom 24339 htpyid 24340 htpyco1 24341 htpyco2 24342 htpycc 24343 isphtpy2d 24350 |
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