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Theorem fiunlem 7715
Description: Lemma for fiun 7716 and f1iun 7717. Formerly part of f1iun 7717. (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 3412 . . . 4 𝑣 ∈ V
2 eqeq1 2741 . . . . 5 (𝑧 = 𝑣 → (𝑧 = 𝐵𝑣 = 𝐵))
32rexbidv 3216 . . . 4 (𝑧 = 𝑣 → (∃𝑥𝐴 𝑧 = 𝐵 ↔ ∃𝑥𝐴 𝑣 = 𝐵))
41, 3elab 3587 . . 3 (𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} ↔ ∃𝑥𝐴 𝑣 = 𝐵)
5 fiun.1 . . . . . 6 (𝑥 = 𝑦𝐵 = 𝐶)
65eqeq2d 2748 . . . . 5 (𝑥 = 𝑦 → (𝑣 = 𝐵𝑣 = 𝐶))
76cbvrexvw 3359 . . . 4 (∃𝑥𝐴 𝑣 = 𝐵 ↔ ∃𝑦𝐴 𝑣 = 𝐶)
8 r19.29 3176 . . . . . . 7 ((∀𝑦𝐴 (𝐵𝐶𝐶𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) → ∃𝑦𝐴 ((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶))
9 sseq12 3928 . . . . . . . . . . . . 13 ((𝑢 = 𝐵𝑣 = 𝐶) → (𝑢𝑣𝐵𝐶))
109ancoms 462 . . . . . . . . . . . 12 ((𝑣 = 𝐶𝑢 = 𝐵) → (𝑢𝑣𝐵𝐶))
11 sseq12 3928 . . . . . . . . . . . 12 ((𝑣 = 𝐶𝑢 = 𝐵) → (𝑣𝑢𝐶𝐵))
1210, 11orbi12d 919 . . . . . . . . . . 11 ((𝑣 = 𝐶𝑢 = 𝐵) → ((𝑢𝑣𝑣𝑢) ↔ (𝐵𝐶𝐶𝐵)))
1312biimprcd 253 . . . . . . . . . 10 ((𝐵𝐶𝐶𝐵) → ((𝑣 = 𝐶𝑢 = 𝐵) → (𝑢𝑣𝑣𝑢)))
1413expdimp 456 . . . . . . . . 9 (((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶) → (𝑢 = 𝐵 → (𝑢𝑣𝑣𝑢)))
1514rexlimivw 3201 . . . . . . . 8 (∃𝑦𝐴 ((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶) → (𝑢 = 𝐵 → (𝑢𝑣𝑣𝑢)))
1615imp 410 . . . . . . 7 ((∃𝑦𝐴 ((𝐵𝐶𝐶𝐵) ∧ 𝑣 = 𝐶) ∧ 𝑢 = 𝐵) → (𝑢𝑣𝑣𝑢))
178, 16sylan 583 . . . . . 6 (((∀𝑦𝐴 (𝐵𝐶𝐶𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) ∧ 𝑢 = 𝐵) → (𝑢𝑣𝑣𝑢))
1817an32s 652 . . . . 5 (((∀𝑦𝐴 (𝐵𝐶𝐶𝐵) ∧ 𝑢 = 𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) → (𝑢𝑣𝑣𝑢))
1918adantlll 718 . . . 4 ((((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) ∧ ∃𝑦𝐴 𝑣 = 𝐶) → (𝑢𝑣𝑣𝑢))
207, 19sylan2b 597 . . 3 ((((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) ∧ ∃𝑥𝐴 𝑣 = 𝐵) → (𝑢𝑣𝑣𝑢))
214, 20sylan2b 597 . 2 ((((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) ∧ 𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵}) → (𝑢𝑣𝑣𝑢))
2221ralrimiva 3105 1 (((𝐵:𝐷𝑆 ∧ ∀𝑦𝐴 (𝐵𝐶𝐶𝐵)) ∧ 𝑢 = 𝐵) → ∀𝑣 ∈ {𝑧 ∣ ∃𝑥𝐴 𝑧 = 𝐵} (𝑢𝑣𝑣𝑢))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 847   = wceq 1543  wcel 2110  {cab 2714  wral 3061  wrex 3062  wss 3866  wf 6376
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rex 3067  df-v 3410  df-in 3873  df-ss 3883
This theorem is referenced by:  fiun  7716  f1iun  7717
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