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Theorem fiunlem 7875
Description: Lemma for fiun 7876 and f1iun 7877. Formerly part of f1iun 7877. (Contributed by AV, 6-Oct-2023.)
Hypothesis
Ref Expression
fiun.1 (𝑥 = 𝑦𝐵 = 𝐶)
Assertion
Ref Expression
fiunlem (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
Distinct variable groups:   𝑣,𝐴,𝑥,𝑧   𝑦,𝐴,𝑣   𝑣,𝐵,𝑦   𝑧,𝐵   𝑣,𝐶,𝑥   𝑣,𝐷   𝑣,𝑆   𝑣,𝑢,𝑦   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑢)   𝐵(𝑥,𝑢)   𝐶(𝑦,𝑧,𝑢)   𝐷(𝑥,𝑦,𝑧,𝑢)   𝑆(𝑥,𝑦,𝑧,𝑢)

Proof of Theorem fiunlem
StepHypRef Expression
1 vex 3448 . . . 4 𝑣 ∈ V
2 eqeq1 2737 . . . . 5 (𝑧 = 𝑣 → (𝑧 = 𝐵𝑣 = 𝐵))
32rexbidv 3172 . . . 4 (𝑧 = 𝑣 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑣 = 𝐵))
41, 3elab 3631 . . 3 (𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑣 = 𝐵)
5 fiun.1 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
65eqeq2d 2744 . . . . 5 (𝑥 = 𝑦 → (𝑣 = 𝐵𝑣 = 𝐶))
76cbvrexvw 3225 . . . 4 (∃𝑥𝐴 𝑣 = 𝐵 ↔ ∃𝑦𝐴 𝑣 = 𝐶)
8 r19.29 3114 . . . . . . 7 ((∀𝑦𝐴 (𝐵𝐶𝐶𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) → ∃𝑦𝐴 ((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶))
9 sseq12 3972 . . . . . . . . . . . . 13 ((𝑢 = 𝐵𝑣 = 𝐶) → (𝑢𝑣𝐵𝐶))
109ancoms 460 . . . . . . . . . . . 12 ((𝑣 = 𝐶𝑢 = 𝐵) → (𝑢𝑣𝐵𝐶))
11 sseq12 3972 . . . . . . . . . . . 12 ((𝑣 = 𝐶𝑢 = 𝐵) → (𝑣𝑢𝐶𝐵))
1210, 11orbi12d 918 . . . . . . . . . . 11 ((𝑣 = 𝐶𝑢 = 𝐵) → ((𝑢𝑣𝑣𝑢) ↔ (𝐵𝐶𝐶𝐵)))
1312biimprcd 250 . . . . . . . . . 10 ((𝐵𝐶𝐶𝐵) → ((𝑣 = 𝐶𝑢 = 𝐵) → (𝑢𝑣𝑣𝑢)))
1413expdimp 454 . . . . . . . . 9 (((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶) → (𝑢 = 𝐵 → (𝑢𝑣𝑣𝑢)))
1514rexlimivw 3145 . . . . . . . 8 (∃𝑦𝐴 ((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶) → (𝑢 = 𝐵 → (𝑢𝑣𝑣𝑢)))
1615imp 408 . . . . . . 7 ((∃𝑦𝐴 ((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶) ∧ 𝑢 = 𝐵) → (𝑢𝑣𝑣𝑢))
178, 16sylan 581 . . . . . 6 (((∀𝑦𝐴 (𝐵𝐶𝐶𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) ∧ 𝑢 = 𝐵) → (𝑢𝑣𝑣𝑢))
1817an32s 651 . . . . 5 (((∀𝑦𝐴 (𝐵𝐶𝐶𝐵) ∧ 𝑢 = 𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) → (𝑢𝑣𝑣𝑢))
1918adantlll 717 . . . 4 ((((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) → (𝑢𝑣𝑣𝑢))
207, 19sylan2b 595 . . 3 ((((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) ∧ ∃𝑥𝐴 𝑣 = 𝐵) → (𝑢𝑣𝑣𝑢))
214, 20sylan2b 595 . 2 ((((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → (𝑢𝑣𝑣𝑢))
2221ralrimiva 3140 1 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846   = wceq 1542  wcel 2107  {cab 2710  wral 3061  wrex 3070  wss 3911  wf 6493
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-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-v 3446  df-in 3918  df-ss 3928
This theorem is referenced by:  fiun  7876  f1iun  7877
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