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Theorem f1otrgitv 25970
 Description: Convenient lemma for f1otrg 25971. (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 (𝜑𝐹:𝐵1-1-onto𝑃)
f1otrkg.1 ((𝜑 ∧ (𝑒𝐵𝑓𝐵)) → (𝑒𝐸𝑓) = ((𝐹𝑒)𝐷(𝐹𝑓)))
f1otrkg.2 ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))
f1otrgitv.x (𝜑𝑋𝐵)
f1otrgitv.y (𝜑𝑌𝐵)
f1otrgitv.z (𝜑𝑍𝐵)
Assertion
Ref Expression
f1otrgitv (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
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
StepHypRef Expression
1 f1otrkg.2 . . 3 ((𝜑 ∧ (𝑒𝐵𝑓𝐵𝑔𝐵)) → (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))
21ralrimivvva 3120 . 2 (𝜑 → ∀𝑒𝐵𝑓𝐵𝑔𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))))
3 f1otrgitv.x . . 3 (𝜑𝑋𝐵)
4 f1otrgitv.y . . 3 (𝜑𝑌𝐵)
5 f1otrgitv.z . . 3 (𝜑𝑍𝐵)
6 oveq1 6799 . . . . . 6 (𝑒 = 𝑋 → (𝑒𝐽𝑓) = (𝑋𝐽𝑓))
76eleq2d 2835 . . . . 5 (𝑒 = 𝑋 → (𝑔 ∈ (𝑒𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑓)))
8 fveq2 6332 . . . . . . 7 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
98oveq1d 6807 . . . . . 6 (𝑒 = 𝑋 → ((𝐹𝑒)𝐼(𝐹𝑓)) = ((𝐹𝑋)𝐼(𝐹𝑓)))
109eleq2d 2835 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓)) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓))))
117, 10bibi12d 334 . . . 4 (𝑒 = 𝑋 → ((𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓)))))
12 oveq2 6800 . . . . . 6 (𝑓 = 𝑌 → (𝑋𝐽𝑓) = (𝑋𝐽𝑌))
1312eleq2d 2835 . . . . 5 (𝑓 = 𝑌 → (𝑔 ∈ (𝑋𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑌)))
14 fveq2 6332 . . . . . . 7 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1514oveq2d 6808 . . . . . 6 (𝑓 = 𝑌 → ((𝐹𝑋)𝐼(𝐹𝑓)) = ((𝐹𝑋)𝐼(𝐹𝑌)))
1615eleq2d 2835 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓)) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
1713, 16bibi12d 334 . . . 4 (𝑓 = 𝑌 → ((𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
18 eleq1 2837 . . . . 5 (𝑔 = 𝑍 → (𝑔 ∈ (𝑋𝐽𝑌) ↔ 𝑍 ∈ (𝑋𝐽𝑌)))
19 fveq2 6332 . . . . . 6 (𝑔 = 𝑍 → (𝐹𝑔) = (𝐹𝑍))
2019eleq1d 2834 . . . . 5 (𝑔 = 𝑍 → ((𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
2118, 20bibi12d 334 . . . 4 (𝑔 = 𝑍 → ((𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))) ↔ (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
2211, 17, 21rspc3v 3473 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑒𝐵𝑓𝐵𝑔𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
233, 4, 5, 22syl3anc 1475 . 2 (𝜑 → (∀𝑒𝐵𝑓𝐵𝑔𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
242, 23mpd 15 1 (𝜑 → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382   ∧ w3a 1070   = wceq 1630   ∈ wcel 2144  ∀wral 3060  –1-1-onto→wf1o 6030  ‘cfv 6031  (class class class)co 6792  Basecbs 16063  distcds 16157  Itvcitv 25555 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039  df-ov 6795 This theorem is referenced by:  f1otrg  25971  f1otrge  25972
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