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Theorem isthincd2lem1 47647
Description: Lemma for isthincd2 47658 and thincmo2 47648. (Contributed by Zhi Wang, 17-Sep-2024.)
Hypotheses
Ref Expression
isthincd2lem1.1 (𝜑𝑋𝐵)
isthincd2lem1.2 (𝜑𝑌𝐵)
isthincd2lem1.3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
isthincd2lem1.4 (𝜑𝐺 ∈ (𝑋𝐻𝑌))
isthincd2lem1.5 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
Assertion
Ref Expression
isthincd2lem1 (𝜑𝐹 = 𝐺)
Distinct variable groups:   𝑦,𝐵,𝑥   𝑓,𝐻,𝑥,𝑦   𝑓,𝑋
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐵(𝑓)   𝐹(𝑥,𝑦,𝑓)   𝐺(𝑥,𝑦,𝑓)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦,𝑓)

Proof of Theorem isthincd2lem1
Dummy variables 𝑤 𝑧 𝑘 𝑙 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isthincd2lem1.5 . . . . 5 (𝜑 → ∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
2 oveq1 7416 . . . . . . . 8 (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
32eleq2d 2820 . . . . . . 7 (𝑥 = 𝑧 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑦)))
43mobidv 2544 . . . . . 6 (𝑥 = 𝑧 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦)))
5 oveq2 7417 . . . . . . . 8 (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
65eleq2d 2820 . . . . . . 7 (𝑦 = 𝑤 → (𝑓 ∈ (𝑧𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑤)))
76mobidv 2544 . . . . . 6 (𝑦 = 𝑤 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤)))
84, 7cbvral2vw 3239 . . . . 5 (∀𝑥𝐵𝑦𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑧𝐵𝑤𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
91, 8sylib 217 . . . 4 (𝜑 → ∀𝑧𝐵𝑤𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
10 oveq1 7416 . . . . . . 7 (𝑧 = 𝑋 → (𝑧𝐻𝑤) = (𝑋𝐻𝑤))
1110eleq2d 2820 . . . . . 6 (𝑧 = 𝑋 → (𝑓 ∈ (𝑧𝐻𝑤) ↔ 𝑓 ∈ (𝑋𝐻𝑤)))
1211mobidv 2544 . . . . 5 (𝑧 = 𝑋 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤)))
13 nfv 1918 . . . . . . 7 𝑘 𝑓 ∈ (𝑋𝐻𝑤)
14 nfv 1918 . . . . . . 7 𝑓 𝑘 ∈ (𝑋𝐻𝑤)
15 eleq1w 2817 . . . . . . 7 (𝑓 = 𝑘 → (𝑓 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑤)))
1613, 14, 15cbvmow 2598 . . . . . 6 (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤))
17 oveq2 7417 . . . . . . . 8 (𝑤 = 𝑌 → (𝑋𝐻𝑤) = (𝑋𝐻𝑌))
1817eleq2d 2820 . . . . . . 7 (𝑤 = 𝑌 → (𝑘 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑌)))
1918mobidv 2544 . . . . . 6 (𝑤 = 𝑌 → (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
2016, 19bitrid 283 . . . . 5 (𝑤 = 𝑌 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
21 isthincd2lem1.1 . . . . 5 (𝜑𝑋𝐵)
22 eqidd 2734 . . . . 5 ((𝜑𝑧 = 𝑋) → 𝐵 = 𝐵)
23 isthincd2lem1.2 . . . . 5 (𝜑𝑌𝐵)
2412, 20, 21, 22, 23rspc2vd 3945 . . . 4 (𝜑 → (∀𝑧𝐵𝑤𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
259, 24mpd 15 . . 3 (𝜑 → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌))
26 moel 3399 . . 3 (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌) ↔ ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
2725, 26sylib 217 . 2 (𝜑 → ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
28 eqeq1 2737 . . 3 (𝑘 = 𝐹 → (𝑘 = 𝑙𝐹 = 𝑙))
29 eqeq2 2745 . . 3 (𝑙 = 𝐺 → (𝐹 = 𝑙𝐹 = 𝐺))
30 isthincd2lem1.3 . . 3 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
31 eqidd 2734 . . 3 ((𝜑𝑘 = 𝐹) → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
32 isthincd2lem1.4 . . 3 (𝜑𝐺 ∈ (𝑋𝐻𝑌))
3328, 29, 30, 31, 32rspc2vd 3945 . 2 (𝜑 → (∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙𝐹 = 𝐺))
3427, 33mpd 15 1 (𝜑𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  ∃*wmo 2533  wral 3062  (class class class)co 7409
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-10 2138  ax-11 2155  ax-12 2172  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-nf 1787  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  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:  thincmo2  47648  isthincd2  47658
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