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Mirrors > Home > MPE Home > Th. List > phtpcrel | Structured version Visualization version GIF version |
Description: The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.) |
Ref | Expression |
---|---|
phtpcrel | ⊢ Rel ( ≃ph‘𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-phtpc 23203 | . 2 ⊢ ≃ph = (𝑥 ∈ Top ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)}) | |
2 | 1 | relmptopab 7162 | 1 ⊢ Rel ( ≃ph‘𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ≠ wne 2969 ⊆ wss 3792 ∅c0 4141 {cpr 4400 Rel wrel 5362 ‘cfv 6137 (class class class)co 6924 Topctop 21109 Cn ccn 21440 IIcii 23090 PHtpycphtpy 23179 ≃phcphtpc 23180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5019 ax-nul 5027 ax-pow 5079 ax-pr 5140 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-opab 4951 df-mpt 4968 df-id 5263 df-xp 5363 df-rel 5364 df-cnv 5365 df-co 5366 df-dm 5367 df-rn 5368 df-res 5369 df-ima 5370 df-iota 6101 df-fun 6139 df-fv 6145 df-phtpc 23203 |
This theorem is referenced by: phtpcer 23206 |
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