Theorem f1otrgitv 25550

Description: Convenient lemma for f1otrg 25551. (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 2955	. 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 6556	. . . . . 6 ⊢ (𝑒 = 𝑋 → (𝑒𝐽𝑓) = (𝑋𝐽𝑓))
7	6	eleq2d 2673	. . . . 5 ⊢ (𝑒 = 𝑋 → (𝑔 ∈ (𝑒𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑓)))
8		fveq2 6103	. . . . . . 7 ⊢ (𝑒 = 𝑋 → (𝐹‘𝑒) = (𝐹‘𝑋))
9	8	oveq1d 6564	. . . . . 6 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑒)𝐼(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐼(𝐹‘𝑓)))
10	9	eleq2d 2673	. . . . 5 ⊢ (𝑒 = 𝑋 → ((𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓)) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓))))
11	7, 10	bibi12d 334	. . . 4 ⊢ (𝑒 = 𝑋 → ((𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑒)𝐼(𝐹‘𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓)))))
12		oveq2 6557	. . . . . 6 ⊢ (𝑓 = 𝑌 → (𝑋𝐽𝑓) = (𝑋𝐽𝑌))
13	12	eleq2d 2673	. . . . 5 ⊢ (𝑓 = 𝑌 → (𝑔 ∈ (𝑋𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑌)))
14		fveq2 6103	. . . . . . 7 ⊢ (𝑓 = 𝑌 → (𝐹‘𝑓) = (𝐹‘𝑌))
15	14	oveq2d 6565	. . . . . 6 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑋)𝐼(𝐹‘𝑓)) = ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))
16	15	eleq2d 2673	. . . . 5 ⊢ (𝑓 = 𝑌 → ((𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓)) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))))
17	13, 16	bibi12d 334	. . . 4 ⊢ (𝑓 = 𝑌 → ((𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)))))
18		eleq1 2676	. . . . 5 ⊢ (𝑔 = 𝑍 → (𝑔 ∈ (𝑋𝐽𝑌) ↔ 𝑍 ∈ (𝑋𝐽𝑌)))
19		fveq2 6103	. . . . . 6 ⊢ (𝑔 = 𝑍 → (𝐹‘𝑔) = (𝐹‘𝑍))
20	19	eleq1d 2672	. . . . 5 ⊢ (𝑔 = 𝑍 → ((𝐹‘𝑔) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌)) ↔ (𝐹‘𝑍) ∈ ((𝐹‘𝑋)𝐼(𝐹‘𝑌))))
21	18, 20	bibi12d 334	. . . 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 3296	. . 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 1318	. 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 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  distcds 15777  Itvcitv 25135

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552

This theorem is referenced by:  f1otrg  25551  f1otrge  25552