Theorem f1omoOLD 50001

Description: Obsolete version of f1omo 50000 as of 24-Nov-2025. (Contributed by Zhi Wang, 19-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypothesis

Ref	Expression
f1omo.1	⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))

Assertion

Ref	Expression
f1omoOLD	⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝑋   𝜑,𝑦

Proof of Theorem f1omoOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1oex 8486	. . . 4 ⊢ 1o ∈ V
2		eqid 2761	. . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
3	1, 2	fvconst0ci 49998	. . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4		mo0 49923	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5		el1o 8503	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
6		el1o 8503	. . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
7		eqtr3 2783	. . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥)
8	5, 6, 7	syl2anb 610	. . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)
9	8	gen2 1829	. . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)
10		eleq1w 2844	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o))
11	10	mo4 2592	. . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥))
12	9, 11	mpbir 234	. . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o
13		eleq2w2 2757	. . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o))
14	13	mobidv 2575	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o))
15	12, 14	mpbiri 261	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
16	4, 15	jaoi 871	. . 3 ⊢ ((((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o) → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
17	3, 16	ax-mp 5	. 2 ⊢ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)
18		f1omo.1	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝐴 × {1o}))
19	18	fveq1d 6887	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
20	19	eleq2d 2847	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
21	20	mobidv 2575	. 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
22	17, 21	mpbiri 261	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∃*wmo 2563  ∅c0 4279  {csn 4584   × cxp 5649  ‘cfv 6538  1_oc1o 8469

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-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  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-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-1o 8476

This theorem is used by: (None)