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Theorem f1otrgds 27853
Description: Convenient lemma for f1otrg 27855. (Contributed by Thierry Arnoux, 19-Mar-2019.)
Hypotheses
Ref Expression
f1otrkg.p 𝑃 = (Baseβ€˜πΊ)
f1otrkg.d 𝐷 = (distβ€˜πΊ)
f1otrkg.i 𝐼 = (Itvβ€˜πΊ)
f1otrkg.b 𝐡 = (Baseβ€˜π»)
f1otrkg.e 𝐸 = (distβ€˜π»)
f1otrkg.j 𝐽 = (Itvβ€˜π»)
f1otrkg.f (πœ‘ β†’ 𝐹:𝐡–1-1-onto→𝑃)
f1otrkg.1 ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡)) β†’ (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))
f1otrkg.2 ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))
f1otrgitv.x (πœ‘ β†’ 𝑋 ∈ 𝐡)
f1otrgitv.y (πœ‘ β†’ π‘Œ ∈ 𝐡)
Assertion
Ref Expression
f1otrgds (πœ‘ β†’ (π‘‹πΈπ‘Œ) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ)))
Distinct variable groups:   𝑒,𝑓,𝑔,𝐡   𝐷,𝑒,𝑓   𝑒,𝐸,𝑓   𝑒,𝐹,𝑓,𝑔   𝑒,𝐼,𝑓,𝑔   𝑒,𝐽,𝑓,𝑔   𝑒,𝑋,𝑓,𝑔   πœ‘,𝑒,𝑓,𝑔   𝑓,π‘Œ,𝑔
Allowed substitution hints:   𝐷(𝑔)   𝑃(𝑒,𝑓,𝑔)   𝐸(𝑔)   𝐺(𝑒,𝑓,𝑔)   𝐻(𝑒,𝑓,𝑔)   π‘Œ(𝑒)

Proof of Theorem f1otrgds
StepHypRef Expression
1 f1otrkg.1 . . 3 ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡)) β†’ (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))
21ralrimivva 3198 . 2 (πœ‘ β†’ βˆ€π‘’ ∈ 𝐡 βˆ€π‘“ ∈ 𝐡 (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)))
3 f1otrgitv.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
4 f1otrgitv.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
5 oveq1 7369 . . . . 5 (𝑒 = 𝑋 β†’ (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 6847 . . . . . 6 (𝑒 = 𝑋 β†’ (πΉβ€˜π‘’) = (πΉβ€˜π‘‹))
76oveq1d 7377 . . . . 5 (𝑒 = 𝑋 β†’ ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“)))
85, 7eqeq12d 2753 . . . 4 (𝑒 = 𝑋 β†’ ((𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)) ↔ (𝑋𝐸𝑓) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“))))
9 oveq2 7370 . . . . 5 (𝑓 = π‘Œ β†’ (𝑋𝐸𝑓) = (π‘‹πΈπ‘Œ))
10 fveq2 6847 . . . . . 6 (𝑓 = π‘Œ β†’ (πΉβ€˜π‘“) = (πΉβ€˜π‘Œ))
1110oveq2d 7378 . . . . 5 (𝑓 = π‘Œ β†’ ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“)) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ)))
129, 11eqeq12d 2753 . . . 4 (𝑓 = π‘Œ β†’ ((𝑋𝐸𝑓) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“)) ↔ (π‘‹πΈπ‘Œ) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ))))
138, 12rspc2v 3593 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡) β†’ (βˆ€π‘’ ∈ 𝐡 βˆ€π‘“ ∈ 𝐡 (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)) β†’ (π‘‹πΈπ‘Œ) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ))))
143, 4, 13syl2anc 585 . 2 (πœ‘ β†’ (βˆ€π‘’ ∈ 𝐡 βˆ€π‘“ ∈ 𝐡 (𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)) β†’ (π‘‹πΈπ‘Œ) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ))))
152, 14mpd 15 1 (πœ‘ β†’ (π‘‹πΈπ‘Œ) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  β€“1-1-ontoβ†’wf1o 6500  β€˜cfv 6501  (class class class)co 7362  Basecbs 17090  distcds 17149  Itvcitv 27417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-iota 6453  df-fv 6509  df-ov 7365
This theorem is referenced by:  f1otrg  27855
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