Theorem eupthf1o 30862

Description: The 𝐹 function in an Eulerian path is a bijection from a half-open range of nonnegative integers to the set of edges. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 18-Feb-2021.)

Hypothesis

Ref	Expression
eupths.i	⊢ 𝐼 = (iEdg‘𝐺)

Assertion

Ref	Expression
eupthf1o	⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼)

Proof of Theorem eupthf1o

Step	Hyp	Ref	Expression
1		eupths.i	. . 3 ⊢ 𝐼 = (iEdg‘𝐺)
2	1	eupthi 30861	. 2 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → (𝐹(Walks‘𝐺)𝑃 ∧ 𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼))
3	2	simprd 501	1 ⊢ (𝐹(EulerPaths‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))⟶1-1ontodom 𝐼)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   class class class wbr 5108  dom cdm 5658  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  0cc0 11239  ..^cfzo 13828  ♯chash 14515  iEdgciedg 29632  Walkscwlks 30234  EulerPathsceupth 30855

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-wlks 30237  df-trls 30332  df-eupth 30856

This theorem is used by:  eupthfi  30863  eupthvdres  30893  eucrctshift  30901