Theorem f1otrgitv 26664

Description: Convenient lemma for f1otrg 26665. (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)
f1otrgitv.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
f1otrgitv	⊢ (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(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   𝑔,𝑍

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

Proof of Theorem f1otrgitv

Step	Hyp	Ref	Expression
1		f1otrkg.2	. . 3 ⊢ ((𝜑 ∧ (𝑒 ∈ 𝐵 ∧ 𝑓 ∈ 𝐵 ∧ 𝑔 ∈ 𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))
2	1	ralrimivvva 3157	. 2 ⊢ (𝜑 → ∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))))
3		f1otrgitv.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
4		f1otrgitv.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		f1otrgitv.z	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
6		oveq1 7142	. . . . . 6 ⊢ (𝑒 = 𝑋 → (𝑒𝐽𝑓) = (𝑋𝐽𝑓))
7	6	eleq2d 2875	. . . . 5 ⊢ (𝑒 = 𝑋 → (𝑔 ∈ (𝑒𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑓)))
8		fveq2 6645	. . . . . . 7 ⊢ (𝑒 = 𝑋 → (𝐹‘𝑒) = (𝐹‘𝑋))
9	8	oveq1d 7150	. . . . . 6 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑒)𝐼(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐼(𝐹‘𝑓)))
10	9	eleq2d 2875	. . . . 5 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓))))
11	7, 10	bibi12d 349	. . . 4 ⊢ (𝑒 = 𝑋 → ((𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓)))))
12		oveq2 7143	. . . . . 6 ⊢ (𝑓 = 𝑌 → (𝑋𝐽𝑓) = (𝑋𝐽𝑌))
13	12	eleq2d 2875	. . . . 5 ⊢ (𝑓 = 𝑌 → (𝑔 ∈ (𝑋𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑌)))
14		fveq2 6645	. . . . . . 7 ⊢ (𝑓 = 𝑌 → (𝐹‘𝑓) = (𝐹‘𝑌))
15	14	oveq2d 7151	. . . . . 6 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑋)𝐼(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))
16	15	eleq2d 2875	. . . . 5 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓)) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))))
17	13, 16	bibi12d 349	. . . 4 ⊢ (𝑓 = 𝑌 → ((𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))))
18		eleq1 2877	. . . . 5 ⊢ (𝑔 = 𝑍 → (𝑔 ∈ (𝑋𝐽𝑌) ↔ 𝑍 ∈ (𝑋𝐽𝑌)))
19		fveq2 6645	. . . . . 6 ⊢ (𝑔 = 𝑍 → (𝐹‘𝑔) = (𝐹‘𝑍))
20	19	eleq1d 2874	. . . . 5 ⊢ (𝑔 = 𝑍 → ((𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))))
21	18, 20	bibi12d 349	. . . 4 ⊢ (𝑔 = 𝑍 → ((𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))) ↔ (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))))
22	11, 17, 21	rspc3v 3584	. . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))) → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))))
23	3, 4, 5, 22	syl3anc 1368	. 2 ⊢ (𝜑 → (∀𝑒 ∈ 𝐵 ∀𝑓 ∈ 𝐵 ∀𝑔 ∈ 𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))) → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))))
24	2, 23	mpd 15	1 ⊢ (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  distcds 16566  Itvcitv 26230

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-v 3443  df-un 3886  df-in 3888  df-ss 3898  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-iota 6283  df-fv 6332  df-ov 7138

This theorem is referenced by:  f1otrg  26665  f1otrge  26666