![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 24739 | . 2 ⊢ ≃ph = (𝑥 ∈ Top ↦ {〈𝑓, 𝑔〉 ∣ ({𝑓, 𝑔} ⊆ (II Cn 𝑥) ∧ (𝑓(PHtpy‘𝑥)𝑔) ≠ ∅)}) | |
2 | 1 | relmptopab 7660 | 1 ⊢ Rel ( ≃ph‘𝐽) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ≠ wne 2939 ⊆ wss 3948 ∅c0 4322 {cpr 4630 Rel wrel 5681 ‘cfv 6543 (class class class)co 7412 Topctop 22616 Cn ccn 22949 IIcii 24616 PHtpycphtpy 24715 ≃phcphtpc 24716 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fv 6551 df-phtpc 24739 |
This theorem is referenced by: phtpcer 24742 |
Copyright terms: Public domain | W3C validator |