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Theorem f1otrgds 28701
Description: Convenient lemma for f1otrg 28703. (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 7433 . . . . 5 (𝑒 = 𝑋 β†’ (𝑒𝐸𝑓) = (𝑋𝐸𝑓))
6 fveq2 6902 . . . . . 6 (𝑒 = 𝑋 β†’ (πΉβ€˜π‘’) = (πΉβ€˜π‘‹))
76oveq1d 7441 . . . . 5 (𝑒 = 𝑋 β†’ ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“)))
85, 7eqeq12d 2744 . . . 4 (𝑒 = 𝑋 β†’ ((𝑒𝐸𝑓) = ((πΉβ€˜π‘’)𝐷(πΉβ€˜π‘“)) ↔ (𝑋𝐸𝑓) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“))))
9 oveq2 7434 . . . . 5 (𝑓 = π‘Œ β†’ (𝑋𝐸𝑓) = (π‘‹πΈπ‘Œ))
10 fveq2 6902 . . . . . 6 (𝑓 = π‘Œ β†’ (πΉβ€˜π‘“) = (πΉβ€˜π‘Œ))
1110oveq2d 7442 . . . . 5 (𝑓 = π‘Œ β†’ ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘“)) = ((πΉβ€˜π‘‹)𝐷(πΉβ€˜π‘Œ)))
129, 11eqeq12d 2744 . . . 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 3622 . . 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 582 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  (class class class)co 7426  Basecbs 17189  distcds 17251  Itvcitv 28265
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-ov 7429
This theorem is referenced by:  f1otrg  28703
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