Theorem initopropdlemlem 50165

Description: Lemma for initopropdlem 50166, termopropdlem 50167, and zeroopropdlem 50168. (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 3927	. . . . . 6 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑌)
4	1, 3	nsyl 141	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑋)
5		initopropdlemlem.1	. . . . . . . 8 ⊢ 𝐹 Fn 𝑋
6	5	fndmi 6636	. . . . . . 7 ⊢ dom 𝐹 = 𝑋
7	6	eleq2i 2852	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑋)
8		ndmfv 6910	. . . . . 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 486	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = ∅)
12		initopropdlemlem.4	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)
13	11, 12	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
14	10	adantr 486	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = ∅)
15	6	eleq2i 2852	. . . . 5 ⊢ (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝑋)
16		ndmfv 6910	. . . . 5 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
17	15, 16	sylnbir 334	. . . 4 ⊢ (¬ 𝐵 ∈ 𝑋 → (𝐹‘𝐵) = ∅)
18	17	adantl 487	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)
19	14, 18	eqtr4d 2798	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
20	13, 19	pm2.61dan 825	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ∅c0 4279  dom cdm 5655   Fn wfn 6528  ‘cfv 6533

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-nul 5263  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5665  df-iota 6489  df-fn 6536  df-fv 6541

This theorem is used by:  initopropdlem  50166  termopropdlem  50167  zeroopropdlem  50168