Theorem isthincd2lem1 49994

Description: Lemma for isthincd2 50006 and thincmo2 49995. (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 7392	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
3	2	eleq2d 2842	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑦)))
4	3	mobidv 2570	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦)))
5		oveq2 7393	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
6	5	eleq2d 2842	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑓 ∈ (𝑧𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑤)))
7	6	mobidv 2570	. . . . . 6 ⊢ (𝑦 = 𝑤 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤)))
8	4, 7	cbvral2vw 3238	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
9	1, 8	sylib 220	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
10		oveq1 7392	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧𝐻𝑤) = (𝑋𝐻𝑤))
11	10	eleq2d 2842	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑓 ∈ (𝑧𝐻𝑤) ↔ 𝑓 ∈ (𝑋𝐻𝑤)))
12	11	mobidv 2570	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤)))
13		nfv 1928	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓 ∈ (𝑋𝐻𝑤)
14		nfv 1928	. . . . . . 7 ⊢ Ⅎ𝑓 𝑘 ∈ (𝑋𝐻𝑤)
15		eleq1w 2839	. . . . . . 7 ⊢ (𝑓 = 𝑘 → (𝑓 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑤)))
16	13, 14, 15	cbvmow 2624	. . . . . 6 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤))
17		oveq2 7393	. . . . . . . 8 ⊢ (𝑤 = 𝑌 → (𝑋𝐻𝑤) = (𝑋𝐻𝑌))
18	17	eleq2d 2842	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (𝑘 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑌)))
19	18	mobidv 2570	. . . . . 6 ⊢ (𝑤 = 𝑌 → (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
20	16, 19	bitrid 285	. . . . 5 ⊢ (𝑤 = 𝑌 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
21		isthincd2lem1.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		eqidd 2757	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑋) → 𝐵 = 𝐵)
23		isthincd2lem1.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
24	12, 20, 21, 22, 23	rspc2vd 3895	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
25	9, 24	mpd 15	. . 3 ⊢ (𝜑 → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌))
26		moel 3381	. . 3 ⊢ (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌) ↔ ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
27	25, 26	sylib 220	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
28		eqeq1 2760	. . 3 ⊢ (𝑘 = 𝐹 → (𝑘 = 𝑙 ↔ 𝐹 = 𝑙))
29		eqeq2 2768	. . 3 ⊢ (𝑙 = 𝐺 → (𝐹 = 𝑙 ↔ 𝐹 = 𝐺))
30		isthincd2lem1.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
31		eqidd 2757	. . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐹) → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
32		isthincd2lem1.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
33	28, 29, 30, 31, 32	rspc2vd 3895	. 2 ⊢ (𝜑 → (∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙 → 𝐹 = 𝐺))
34	27, 33	mpd 15	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1554   ∈ wcel 2136  ∃*wmo 2558  ∀wral 3070  (class class class)co 7385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-10 2169  ax-11 2185  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-3an 1097  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-mo 2560  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ral 3071  df-rab 3409  df-v 3450  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4281  df-if 4475  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5095  df-iota 6466  df-fv 6518  df-ov 7388

This theorem is referenced by:  thincmo2  49995  isthincd2  50006