Theorem isthincd2lem1 50186

Description: Lemma for isthincd2 50198 and thincmo2 50187. (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 7419	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥𝐻𝑦) = (𝑧𝐻𝑦))
3	2	eleq2d 2849	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (𝑓 ∈ (𝑥𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑦)))
4	3	mobidv 2577	. . . . . 6 ⊢ (𝑥 = 𝑧 → (∃*𝑓 𝑓 ∈ (𝑥𝐻𝑦) ↔ ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑦)))
5		oveq2 7420	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑧𝐻𝑦) = (𝑧𝐻𝑤))
6	5	eleq2d 2849	. . . . . . 7 ⊢ (𝑦 = 𝑤 → (𝑓 ∈ (𝑧𝐻𝑦) ↔ 𝑓 ∈ (𝑧𝐻𝑤)))
7	6	mobidv 2577	. . . . . 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 221	. . . 4 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤))
10		oveq1 7419	. . . . . . 7 ⊢ (𝑧 = 𝑋 → (𝑧𝐻𝑤) = (𝑋𝐻𝑤))
11	10	eleq2d 2849	. . . . . 6 ⊢ (𝑧 = 𝑋 → (𝑓 ∈ (𝑧𝐻𝑤) ↔ 𝑓 ∈ (𝑋𝐻𝑤)))
12	11	mobidv 2577	. . . . 5 ⊢ (𝑧 = 𝑋 → (∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) ↔ ∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤)))
13		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑘 𝑓 ∈ (𝑋𝐻𝑤)
14		nfv 1944	. . . . . . 7 ⊢ Ⅎ𝑓 𝑘 ∈ (𝑋𝐻𝑤)
15		eleq1w 2846	. . . . . . 7 ⊢ (𝑓 = 𝑘 → (𝑓 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑤)))
16	13, 14, 15	cbvmow 2631	. . . . . 6 ⊢ (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤))
17		oveq2 7420	. . . . . . . 8 ⊢ (𝑤 = 𝑌 → (𝑋𝐻𝑤) = (𝑋𝐻𝑌))
18	17	eleq2d 2849	. . . . . . 7 ⊢ (𝑤 = 𝑌 → (𝑘 ∈ (𝑋𝐻𝑤) ↔ 𝑘 ∈ (𝑋𝐻𝑌)))
19	18	mobidv 2577	. . . . . 6 ⊢ (𝑤 = 𝑌 → (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
20	16, 19	bitrid 286	. . . . 5 ⊢ (𝑤 = 𝑌 → (∃*𝑓 𝑓 ∈ (𝑋𝐻𝑤) ↔ ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
21		isthincd2lem1.1	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
22		eqidd 2764	. . . . 5 ⊢ ((𝜑 ∧ 𝑧 = 𝑋) → 𝐵 = 𝐵)
23		isthincd2lem1.2	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
24	12, 20, 21, 22, 23	rspc2vd 3902	. . . 4 ⊢ (𝜑 → (∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 ∃*𝑓 𝑓 ∈ (𝑧𝐻𝑤) → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌)))
25	9, 24	mpd 16	. . 3 ⊢ (𝜑 → ∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌))
26		moel 3389	. . 3 ⊢ (∃*𝑘 𝑘 ∈ (𝑋𝐻𝑌) ↔ ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
27	25, 26	sylib 221	. 2 ⊢ (𝜑 → ∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙)
28		eqeq1 2767	. . 3 ⊢ (𝑘 = 𝐹 → (𝑘 = 𝑙 ↔ 𝐹 = 𝑙))
29		eqeq2 2775	. . 3 ⊢ (𝑙 = 𝐺 → (𝐹 = 𝑙 ↔ 𝐹 = 𝐺))
30		isthincd2lem1.3	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
31		eqidd 2764	. . 3 ⊢ ((𝜑 ∧ 𝑘 = 𝐹) → (𝑋𝐻𝑌) = (𝑋𝐻𝑌))
32		isthincd2lem1.4	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑋𝐻𝑌))
33	28, 29, 30, 31, 32	rspc2vd 3902	. 2 ⊢ (𝜑 → (∀𝑘 ∈ (𝑋𝐻𝑌)∀𝑙 ∈ (𝑋𝐻𝑌)𝑘 = 𝑙 → 𝐹 = 𝐺))
34	27, 33	mpd 16	1 ⊢ (𝜑 → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∃*wmo 2565  ∀wral 3079  (class class class)co 7412

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-iota 6494  df-fv 6546  df-ov 7415

This theorem is referenced by:  thincmo2  50187  isthincd2  50198