Theorem f1omoOLD 49384

Description: Obsolete version of f1omo 49383 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 8405	. . . 4 ⊢ 1o ∈ V
2		eqid 2739	. . . 4 ⊢ ((𝐴 × {1o})‘𝑋) = ((𝐴 × {1o})‘𝑋)
3	1, 2	fvconst0ci 49381	. . 3 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ ∨ ((𝐴 × {1o})‘𝑋) = 1o)
4		mo0 49304	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = ∅ → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
5		el1o 8420	. . . . . . . 8 ⊢ (𝑦 ∈ 1o ↔ 𝑦 = ∅)
6		el1o 8420	. . . . . . . 8 ⊢ (𝑥 ∈ 1o ↔ 𝑥 = ∅)
7		eqtr3 2761	. . . . . . . 8 ⊢ ((𝑦 = ∅ ∧ 𝑥 = ∅) → 𝑦 = 𝑥)
8	5, 6, 7	syl2anb 604	. . . . . . 7 ⊢ ((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)
9	8	gen2 1803	. . . . . 6 ⊢ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥)
10		eleq1w 2822	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 1o ↔ 𝑥 ∈ 1o))
11	10	mo4 2570	. . . . . 6 ⊢ (∃*𝑦 𝑦 ∈ 1o ↔ ∀𝑦∀𝑥((𝑦 ∈ 1o ∧ 𝑥 ∈ 1o) → 𝑦 = 𝑥))
12	9, 11	mpbir 232	. . . . 5 ⊢ ∃*𝑦 𝑦 ∈ 1o
13		eleq2w2 2735	. . . . . 6 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ 𝑦 ∈ 1o))
14	13	mobidv 2553	. . . . 5 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → (∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋) ↔ ∃*𝑦 𝑦 ∈ 1o))
15	12, 14	mpbiri 259	. . . 4 ⊢ (((𝐴 × {1o})‘𝑋) = 1o → ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋))
16	4, 15	jaoi 863	. . 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 6829	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) = ((𝐴 × {1o})‘𝑋))
20	19	eleq2d 2825	. . 3 ⊢ (𝜑 → (𝑦 ∈ (𝐹‘𝑋) ↔ 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
21	20	mobidv 2553	. 2 ⊢ (𝜑 → (∃*𝑦 𝑦 ∈ (𝐹‘𝑋) ↔ ∃*𝑦 𝑦 ∈ ((𝐴 × {1o})‘𝑋)))
22	17, 21	mpbiri 259	1 ⊢ (𝜑 → ∃*𝑦 𝑦 ∈ (𝐹‘𝑋))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853  ∀wal 1545   = wceq 1547   ∈ wcel 2119  ∃*wmo 2541  ∅c0 4261  {csn 4555   × cxp 5616  ‘cfv 6485  1_oc1o 8388

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-1o 8395

This theorem is referenced by: (None)