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Theorem f1otrgitv 26670
Description: Convenient lemma for f1otrg 26671. (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 3187 . 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 7157 . . . . . 6 (𝑒 = 𝑋 → (𝑒𝐽𝑓) = (𝑋𝐽𝑓))
76eleq2d 2901 . . . . 5 (𝑒 = 𝑋 → (𝑔 ∈ (𝑒𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑓)))
8 fveq2 6662 . . . . . . 7 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
98oveq1d 7165 . . . . . 6 (𝑒 = 𝑋 → ((𝐹𝑒)𝐼(𝐹𝑓)) = ((𝐹𝑋)𝐼(𝐹𝑓)))
109eleq2d 2901 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓)) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓))))
117, 10bibi12d 349 . . . 4 (𝑒 = 𝑋 → ((𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓)))))
12 oveq2 7158 . . . . . 6 (𝑓 = 𝑌 → (𝑋𝐽𝑓) = (𝑋𝐽𝑌))
1312eleq2d 2901 . . . . 5 (𝑓 = 𝑌 → (𝑔 ∈ (𝑋𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑌)))
14 fveq2 6662 . . . . . . 7 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1514oveq2d 7166 . . . . . 6 (𝑓 = 𝑌 → ((𝐹𝑋)𝐼(𝐹𝑓)) = ((𝐹𝑋)𝐼(𝐹𝑌)))
1615eleq2d 2901 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓)) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
1713, 16bibi12d 349 . . . 4 (𝑓 = 𝑌 → ((𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
18 eleq1 2903 . . . . 5 (𝑔 = 𝑍 → (𝑔 ∈ (𝑋𝐽𝑌) ↔ 𝑍 ∈ (𝑋𝐽𝑌)))
19 fveq2 6662 . . . . . 6 (𝑔 = 𝑍 → (𝐹𝑔) = (𝐹𝑍))
2019eleq1d 2900 . . . . 5 (𝑔 = 𝑍 → ((𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
2118, 20bibi12d 349 . . . 4 (𝑔 = 𝑍 → ((𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))) ↔ (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
2211, 17, 21rspc3v 3623 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑒𝐵𝑓𝐵𝑔𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
233, 4, 5, 22syl3anc 1368 . 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 209  wa 399  w3a 1084   = wceq 1538  wcel 2115  wral 3133  1-1-ontowf1o 6343  cfv 6344  (class class class)co 7150  Basecbs 16486  distcds 16577  Itvcitv 26236
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-ral 3138  df-v 3483  df-un 3925  df-in 3927  df-ss 3937  df-sn 4552  df-pr 4554  df-op 4558  df-uni 4826  df-br 5054  df-iota 6303  df-fv 6352  df-ov 7153
This theorem is referenced by:  f1otrg  26671  f1otrge  26672
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