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Theorem f1otrgitv 25667
Description: Convenient lemma for f1otrg 25668. (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 2967 . 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 6617 . . . . . 6 (𝑒 = 𝑋 → (𝑒𝐽𝑓) = (𝑋𝐽𝑓))
76eleq2d 2684 . . . . 5 (𝑒 = 𝑋 → (𝑔 ∈ (𝑒𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑓)))
8 fveq2 6153 . . . . . . 7 (𝑒 = 𝑋 → (𝐹𝑒) = (𝐹𝑋))
98oveq1d 6625 . . . . . 6 (𝑒 = 𝑋 → ((𝐹𝑒)𝐼(𝐹𝑓)) = ((𝐹𝑋)𝐼(𝐹𝑓)))
109eleq2d 2684 . . . . 5 (𝑒 = 𝑋 → ((𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓)) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓))))
117, 10bibi12d 335 . . . 4 (𝑒 = 𝑋 → ((𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓)))))
12 oveq2 6618 . . . . . 6 (𝑓 = 𝑌 → (𝑋𝐽𝑓) = (𝑋𝐽𝑌))
1312eleq2d 2684 . . . . 5 (𝑓 = 𝑌 → (𝑔 ∈ (𝑋𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑌)))
14 fveq2 6153 . . . . . . 7 (𝑓 = 𝑌 → (𝐹𝑓) = (𝐹𝑌))
1514oveq2d 6626 . . . . . 6 (𝑓 = 𝑌 → ((𝐹𝑋)𝐼(𝐹𝑓)) = ((𝐹𝑋)𝐼(𝐹𝑌)))
1615eleq2d 2684 . . . . 5 (𝑓 = 𝑌 → ((𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓)) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
1713, 16bibi12d 335 . . . 4 (𝑓 = 𝑌 → ((𝑔 ∈ (𝑋𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑓))) ↔ (𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
18 eleq1 2686 . . . . 5 (𝑔 = 𝑍 → (𝑔 ∈ (𝑋𝐽𝑌) ↔ 𝑍 ∈ (𝑋𝐽𝑌)))
19 fveq2 6153 . . . . . 6 (𝑔 = 𝑍 → (𝐹𝑔) = (𝐹𝑍))
2019eleq1d 2683 . . . . 5 (𝑔 = 𝑍 → ((𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))))
2118, 20bibi12d 335 . . . 4 (𝑔 = 𝑍 → ((𝑔 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑔) ∈ ((𝐹𝑋)𝐼(𝐹𝑌))) ↔ (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
2211, 17, 21rspc3v 3313 . . 3 ((𝑋𝐵𝑌𝐵𝑍𝐵) → (∀𝑒𝐵𝑓𝐵𝑔𝐵 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (𝐹𝑔) ∈ ((𝐹𝑒)𝐼(𝐹𝑓))) → (𝑍 ∈ (𝑋𝐽𝑌) ↔ (𝐹𝑍) ∈ ((𝐹𝑋)𝐼(𝐹𝑌)))))
233, 4, 5, 22syl3anc 1323 . 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 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  1-1-ontowf1o 5851  cfv 5852  (class class class)co 6610  Basecbs 15792  distcds 15882  Itvcitv 25252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613
This theorem is referenced by:  f1otrg  25668  f1otrge  25669
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