Theorem isthincd2lem1 48694

Description: Lemma for isthincd2 48705 and thincmo2 48695. (Contributed by Zhi Wang, 17-Sep-2024.)

Hypotheses

Ref	Expression
isthincd2lem1.1	⊢ (𝜑 → 𝑋 ∈ 𝐵)
isthincd2lem1.2	⊢ (𝜑 → 𝑌 ∈ 𝐵)
isthincd2lem1.3	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
isthincd2lem1.4	⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
isthincd2lem1.5	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))

Assertion

Ref	Expression
isthincd2lem1	⊢ (𝜑 → 𝐹 = 𝐺)

Distinct variable groups:   𝑦,𝐵,𝑥   𝑓,𝐻,𝑥,𝑦   𝑓,𝑋

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑓)   𝐵(𝑓)   𝐹(𝑥,𝑦,𝑓)   𝐺(𝑥,𝑦,𝑓)   𝑋(𝑥,𝑦)   𝑌(𝑥,𝑦,𝑓)

Proof of Theorem isthincd2lem1

Dummy variables 𝑤 𝑧 𝑘 𝑙 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isthincd2lem1.5	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦))
2		oveq1 7455	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
3	2	eleq2d 2830	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑦)))
4	3	mobidv 2552	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦)))
5		oveq2 7456	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
6	5	eleq2d 2830	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑓 ∈ (𝑧𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑤)))
7	6	mobidv 2552	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤)))
8	4, 7	cbvral2vw 3247	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
9	1, 8	sylib 218	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
10		oveq1 7455	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧𝐻𝑤) = (𝑋𝐻𝑤))
11	10	eleq2d 2830	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑓 ∈ (𝑧𝐻𝑤) ↔ 𝑓 ∈ (𝑋𝐻𝑤)))
12	11	mobidv 2552	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤)))
13		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓 ∈ (𝑋𝐻𝑤)
14		nfv 1913	. . . . . . 7 ⊢ Ⅎ𝑓 𝑘 ∈ (𝑋𝐻𝑤)
15		eleq1w 2827	. . . . . . 7 ⊢ (𝑓 = 𝑘 → (𝑓 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑤)))
16	13, 14, 15	cbvmow 2606	. . . . . 6 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤))
17		oveq2 7456	. . . . . . . 8 ⊢ (𝑤 = 𝑌 → (𝑋𝐻𝑤) = (𝑋𝐻𝑌))
18	17	eleq2d 2830	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (𝑘 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑌)))
19	18	mobidv 2552	. . . . . 6 ⊢ (𝑤 = 𝑌 → (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
20	16, 19	bitrid 283	. . . . 5 ⊢ (𝑤 = 𝑌 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
21		isthincd2lem1.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		eqidd 2741	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑋) → 𝐵 = 𝐵)
23		isthincd2lem1.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
24	12, 20, 21, 22, 23	rspc2vd 3972	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
25	9, 24	mpd 15	. . 3 ⊢ (𝜑 → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌))
26		moel 3410	. . 3 ⊢ (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌) ↔ ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
27	25, 26	sylib 218	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
28		eqeq1 2744	. . 3 ⊢ (𝑘 = 𝐹 → (𝑘 = 𝑙 ↔ 𝐹 = 𝑙))
29		eqeq2 2752	. . 3 ⊢ (𝑙 = 𝐺 → (𝐹 = 𝑙 ↔ 𝐹 = 𝐺))
30		isthincd2lem1.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
31		eqidd 2741	. . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐹) → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
32		isthincd2lem1.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
33	28, 29, 30, 31, 32	rspc2vd 3972	. 2 ⊢ (𝜑 → (∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙 → 𝐹 = 𝐺))
34	27, 33	mpd 15	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2108  ∃*wmo 2541  ∀wral 3067  (class class class)co 7448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-iota 6525  df-fv 6581  df-ov 7451

This theorem is referenced by:  thincmo2  48695  isthincd2  48705