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Theorem f1otrgitv 28121
Description: Convenient lemma for f1otrg 28122. (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)
f1otrgitv.z (πœ‘ β†’ 𝑍 ∈ 𝐡)
Assertion
Ref Expression
f1otrgitv (πœ‘ β†’ (𝑍 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(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   𝑔,𝑍
Allowed substitution hints:   𝐷(𝑔)   𝑃(𝑒,𝑓,𝑔)   𝐸(𝑔)   𝐺(𝑒,𝑓,𝑔)   𝐻(𝑒,𝑓,𝑔)   π‘Œ(𝑒)   𝑍(𝑒,𝑓)
Proof of Theorem f1otrgitv
StepHypRef Expression
1 f1otrkg.2 . . 3 ((πœ‘ ∧ (𝑒 ∈ 𝐡 ∧ 𝑓 ∈ 𝐡 ∧ 𝑔 ∈ 𝐡)) β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))
21ralrimivvva 3204 . 2 (πœ‘ β†’ βˆ€π‘’ ∈ 𝐡 βˆ€π‘“ ∈ 𝐡 βˆ€π‘” ∈ 𝐡 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))))
3 f1otrgitv.x . . 3 (πœ‘ β†’ 𝑋 ∈ 𝐡)
4 f1otrgitv.y . . 3 (πœ‘ β†’ π‘Œ ∈ 𝐡)
5 f1otrgitv.z . . 3 (πœ‘ β†’ 𝑍 ∈ 𝐡)
6 oveq1 7416 . . . . . 6 (𝑒 = 𝑋 β†’ (𝑒𝐽𝑓) = (𝑋𝐽𝑓))
76eleq2d 2820 . . . . 5 (𝑒 = 𝑋 β†’ (𝑔 ∈ (𝑒𝐽𝑓) ↔ 𝑔 ∈ (𝑋𝐽𝑓)))
8 fveq2 6892 . . . . . . 7 (𝑒 = 𝑋 β†’ (πΉβ€˜π‘’) = (πΉβ€˜π‘‹))
98oveq1d 7424 . . . . . 6 (𝑒 = 𝑋 β†’ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“)) = ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘“)))
109eleq2d 2820 . . . . 5 (𝑒 = 𝑋 β†’ ((πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“)) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘“))))
117, 10bibi12d 346 . . . 4 (𝑒 = 𝑋 β†’ ((𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))) ↔ (𝑔 ∈ (𝑋𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘“)))))
12 oveq2 7417 . . . . . 6 (𝑓 = π‘Œ β†’ (𝑋𝐽𝑓) = (π‘‹π½π‘Œ))
1312eleq2d 2820 . . . . 5 (𝑓 = π‘Œ β†’ (𝑔 ∈ (𝑋𝐽𝑓) ↔ 𝑔 ∈ (π‘‹π½π‘Œ)))
14 fveq2 6892 . . . . . . 7 (𝑓 = π‘Œ β†’ (πΉβ€˜π‘“) = (πΉβ€˜π‘Œ))
1514oveq2d 7425 . . . . . 6 (𝑓 = π‘Œ β†’ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘“)) = ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ)))
1615eleq2d 2820 . . . . 5 (𝑓 = π‘Œ β†’ ((πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘“)) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ))))
1713, 16bibi12d 346 . . . 4 (𝑓 = π‘Œ β†’ ((𝑔 ∈ (𝑋𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘“))) ↔ (𝑔 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ)))))
18 eleq1 2822 . . . . 5 (𝑔 = 𝑍 β†’ (𝑔 ∈ (π‘‹π½π‘Œ) ↔ 𝑍 ∈ (π‘‹π½π‘Œ)))
19 fveq2 6892 . . . . . 6 (𝑔 = 𝑍 β†’ (πΉβ€˜π‘”) = (πΉβ€˜π‘))
2019eleq1d 2819 . . . . 5 (𝑔 = 𝑍 β†’ ((πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ)) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ))))
2118, 20bibi12d 346 . . . 4 (𝑔 = 𝑍 β†’ ((𝑔 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ))) ↔ (𝑍 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ)))))
2211, 17, 21rspc3v 3628 . . 3 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ 𝐡 ∧ 𝑍 ∈ 𝐡) β†’ (βˆ€π‘’ ∈ 𝐡 βˆ€π‘“ ∈ 𝐡 βˆ€π‘” ∈ 𝐡 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))) β†’ (𝑍 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ)))))
233, 4, 5, 22syl3anc 1372 . 2 (πœ‘ β†’ (βˆ€π‘’ ∈ 𝐡 βˆ€π‘“ ∈ 𝐡 βˆ€π‘” ∈ 𝐡 (𝑔 ∈ (𝑒𝐽𝑓) ↔ (πΉβ€˜π‘”) ∈ ((πΉβ€˜π‘’)𝐼(πΉβ€˜π‘“))) β†’ (𝑍 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(πΉβ€˜π‘Œ)))))
242, 23mpd 15 1 (πœ‘ β†’ (𝑍 ∈ (π‘‹π½π‘Œ) ↔ (πΉβ€˜π‘) ∈ ((πΉβ€˜π‘‹)𝐼(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 3062  β€“1-1-ontoβ†’wf1o 6543  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  Itvcitv 27684
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 2704
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 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412
This theorem is referenced by:  f1otrg  28122  f1otrge  28123
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