Theorem f1otrgds 26657

Description: Convenient lemma for f1otrg 26659. (Contributed by Thierry Arnoux, 19-Mar-2019.)

Hypotheses

Ref	Expression
f1otrkg.p	⊢ 𝑃 = (Base‘𝐺)
f1otrkg.d	⊢ 𝐷 = (dist‘𝐺)
f1otrkg.i	⊢ 𝐼 = (Itv‘𝐺)
f1otrkg.b	⊢ 𝐵 = (Base‘𝐻)
f1otrkg.e	⊢ 𝐸 = (dist‘𝐻)
f1otrkg.j	⊢ 𝐽 = (Itv‘𝐻)
f1otrkg.f	⊢ (𝜑 → 𝐹:𝐵–1-1-onto→𝑃)
f1otrkg.1	⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))
f1otrkg.2	⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))
f1otrgitv.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
f1otrgitv.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
f1otrgds	⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))

Distinct variable groups:   𝑒,𝑓,𝑔,𝐵   𝐷,𝑒,𝑓   𝑒,𝐸,𝑓   𝑒,𝐹,𝑓,𝑔   𝑒,𝐼,𝑓,𝑔   𝑒,𝐽,𝑓,𝑔   𝑒,𝑋,𝑓,𝑔   𝜑,𝑒,𝑓,𝑔   𝑓,𝑌,𝑔

Allowed substitution hints:   𝐷(𝑔)   𝑃(𝑒,𝑓,𝑔)   𝐸(𝑔)   𝐺(𝑒,𝑓,𝑔)   𝐻(𝑒,𝑓,𝑔)   𝑌(𝑒)

Proof of Theorem f1otrgds

Step	Hyp	Ref	Expression
1		f1otrkg.1	. . 3 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵)) → (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))
2	1	ralrimivva 3193	. 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)))
3		f1otrgitv.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		f1otrgitv.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		oveq1 7165	. . . . 5 ⊢ (𝑒 = 𝑋 → (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6		fveq2 6672	. . . . . 6 ⊢ (𝑒 = 𝑋 → (𝐹‘𝑒) = (𝐹‘𝑋))
7	6	oveq1d 7173	. . . . 5 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓)))
8	5, 7	eqeq12d 2839	. . . 4 ⊢ (𝑒 = 𝑋 → ((𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) ↔ (𝑋𝐸𝑓) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓))))
9		oveq2 7166	. . . . 5 ⊢ (𝑓 = 𝑌 → (𝑋𝐸𝑓) = (𝑋𝐸𝑌))
10		fveq2 6672	. . . . . 6 ⊢ (𝑓 = 𝑌 → (𝐹‘𝑓) = (𝐹‘𝑌))
11	10	oveq2d 7174	. . . . 5 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑋)𝐷(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))
12	9, 11	eqeq12d 2839	. . . 4 ⊢ (𝑓 = 𝑌 → ((𝑋𝐸𝑓) = ((𝐹‘𝑋)𝐷(𝐹‘𝑓)) ↔ (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))))
13	8, 12	rspc2v 3635	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))))
14	3, 4, 13	syl2anc 586	. 2 ⊢ (𝜑 → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 (𝑒𝐸𝑓) = ((𝐹‘𝑒)𝐷(𝐹‘𝑓)) → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌))))
15	2, 14	mpd 15	1 ⊢ (𝜑 → (𝑋𝐸𝑌) = ((𝐹‘𝑋)𝐷(𝐹‘𝑌)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114  ∀wral 3140  –1-1-onto→wf1o 6356  ‘cfv 6357  (class class class)co 7158  Basecbs 16485  distcds 16576  Itvcitv 26224

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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-iota 6316  df-fv 6365  df-ov 7161

This theorem is referenced by:  f1otrg  26659