Theorem initopropdlemlem 49201

Description: Lemma for initopropdlem 49202, termopropdlem 49203, and zeroopropdlem 49204. (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 3939	. . . . . 6 ⊢ (𝐴 ∈ 𝑋 → 𝐴 ∈ 𝑌)
4	1, 3	nsyl 140	. . . . 5 ⊢ (𝜑 → ¬ 𝐴 ∈ 𝑋)
5		initopropdlemlem.1	. . . . . . . 8 ⊢ 𝐹 Fn 𝑋
6	5	fndmi 6604	. . . . . . 7 ⊢ dom 𝐹 = 𝑋
7	6	eleq2i 2820	. . . . . 6 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ 𝑋)
8		ndmfv 6875	. . . . . 6 ⊢ (¬ 𝐴 ∈ dom 𝐹 → (𝐹‘𝐴) = ∅)
9	7, 8	sylnbir 331	. . . . 5 ⊢ (¬ 𝐴 ∈ 𝑋 → (𝐹‘𝐴) = ∅)
10	4, 9	syl 17	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) = ∅)
11	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = ∅)
12		initopropdlemlem.4	. . 3 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)
13	11, 12	eqtr4d 2767	. 2 ⊢ ((𝜑 ∧ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
14	10	adantr 480	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = ∅)
15	6	eleq2i 2820	. . . . 5 ⊢ (𝐵 ∈ dom 𝐹 ↔ 𝐵 ∈ 𝑋)
16		ndmfv 6875	. . . . 5 ⊢ (¬ 𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) = ∅)
17	15, 16	sylnbir 331	. . . 4 ⊢ (¬ 𝐵 ∈ 𝑋 → (𝐹‘𝐵) = ∅)
18	17	adantl 481	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐵) = ∅)
19	14, 18	eqtr4d 2767	. 2 ⊢ ((𝜑 ∧ ¬ 𝐵 ∈ 𝑋) → (𝐹‘𝐴) = (𝐹‘𝐵))
20	13, 19	pm2.61dan 812	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐹‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ⊆ wss 3911  ∅c0 4292  dom cdm 5631   Fn wfn 6494  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-dm 5641  df-iota 6452  df-fn 6502  df-fv 6507

This theorem is referenced by:  initopropdlem  49202  termopropdlem  49203  zeroopropdlem  49204