Theorem initopropdlemlem 50017

Description: Lemma for initopropdlem 50018, termopropdlem 50019, and zeroopropdlem 50020. (Contributed by Zhi Wang, 26-Oct-2025.)

Hypotheses

Ref	Expression
initopropdlemlem.1	⊢ 𝐹 Fn 𝑋
initopropdlemlem.2	⊢ (𝜑 → ¬ 𝐴 ∈ 𝑌)
initopropdlemlem.3	⊢ 𝑋 ⊆ 𝑌
initopropdlemlem.4	⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)

Assertion

Ref	Expression
initopropdlemlem	⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))

Proof of Theorem initopropdlemlem

Step	Hyp	Ref	Expression
1		initopropdlemlem.2	. . . . . 6 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑌)
2		initopropdlemlem.3	. . . . . . 7 ⊢ 𝑋 ⊆ 𝑌
3	2	sseli 3933	. . . . . 6 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑌)
4	1, 3	nsyl 141	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑋)
5		initopropdlemlem.1	. . . . . . . 8 ⊢ 𝐹 Fn 𝑋
6	5	fndmi 6639	. . . . . . 7 ⊢ dom 𝐹 = 𝑋
7	6	eleq2i 2855	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑋)
8		ndmfv 6913	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
9	7, 8	sylnbir 334	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝑋 → (𝐹‘𝐴) = ∅)
10	4, 9	syl 18	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = ∅)
11	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = ∅)
12		initopropdlemlem.4	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)
13	11, 12	eqtr4d 2801	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
14	10	adantr 485	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = ∅)
15	6	eleq2i 2855	. . . . 5 ⊢ (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝑋)
16		ndmfv 6913	. . . . 5 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
17	15, 16	sylnbir 334	. . . 4 ⊢ (¬ 𝐵 ∈ 𝑋 → (𝐹‘𝐵) = ∅)
18	17	adantl 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)
19	14, 18	eqtr4d 2801	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
20	13, 19	pm2.61dan 824	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143   ⊆ wss 3905  ∅c0 4286  dom cdm 5661   Fn wfn 6531  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-dm 5671  df-iota 6492  df-fn 6539  df-fv 6544

This theorem is referenced by:  initopropdlem  50018  termopropdlem  50019  zeroopropdlem  50020