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Theorem funcsetcestrclem6 18124

Description: Lemma 6 for funcsetcestrc 18128. (Contributed by AV, 27-Mar-2020.)

**Hypotheses**
Ref	Expression
funcsetcestrc.s	⊢ 𝑆 = (SetCat‘𝑈)
funcsetcestrc.c	⊢ 𝐶 = (Base‘𝑆)
funcsetcestrc.f	⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
funcsetcestrc.u	⊢ (𝜑 → 𝑈 ∈ WUni)
funcsetcestrc.o	⊢ (𝜑 → ω ∈ 𝑈)
funcsetcestrc.g	⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))

**Assertion**
Ref	Expression
funcsetcestrclem6	⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑_m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Distinct variable groups: 𝑥,𝐶 𝑥,𝑋 𝜑,𝑥 𝑦,𝐶,𝑥 𝑦,𝑋 𝑥,𝑌,𝑦 𝜑,𝑦 Allowed substitution hints: 𝑆(𝑥,𝑦) 𝑈(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝐻(𝑥,𝑦)

**Proof of Theorem funcsetcestrclem6**
Step	Hyp	Ref	Expression
1		funcsetcestrc.s	. . . . 5 ⊢ 𝑆 = (SetCat‘𝑈)
2		funcsetcestrc.c	. . . . 5 ⊢ 𝐶 = (Base‘𝑆)
3		funcsetcestrc.f	. . . . 5 ⊢ (𝜑 → 𝐹 = (𝑥 ∈ 𝐶 ↦ {⟨(Base‘ndx), 𝑥⟩}))
4		funcsetcestrc.u	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ WUni)
5		funcsetcestrc.o	. . . . 5 ⊢ (𝜑 → ω ∈ 𝑈)
6		funcsetcestrc.g	. . . . 5 ⊢ (𝜑 → 𝐺 = (𝑥 ∈ 𝐶, 𝑦 ∈ 𝐶 ↦ ( I ↾ (𝑦 ↑_m 𝑥))))
7	1, 2, 3, 4, 5, 6	funcsetcestrclem5 18123	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑_m 𝑋)))
8	7	3adant3 1138	. . 3 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑_m 𝑋)) → (𝑋𝐺𝑌) = ( I ↾ (𝑌 ↑_m 𝑋)))
9	8	fveq1d 6836	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑_m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = (( I ↾ (𝑌 ↑_m 𝑋))‘𝐻))
10		fvresi 7124	. . 3 ⊢ (𝐻 ∈ (𝑌 ↑_m 𝑋) → (( I ↾ (𝑌 ↑_m 𝑋))‘𝐻) = 𝐻)
11	10	3ad2ant3 1141	. 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑_m 𝑋)) → (( I ↾ (𝑌 ↑_m 𝑋))‘𝐻) = 𝐻)
12	9, 11	eqtrd 2775	1 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝐶 ∧ 𝑌 ∈ 𝐶) ∧ 𝐻 ∈ (𝑌 ↑_m 𝑋)) → ((𝑋𝐺𝑌)‘𝐻) = 𝐻)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 {csn 4562 ⟨cop 4568 ↦ cmpt 5160 I cid 5519 ↾ cres 5627 ‘cfv 6492 (class class class)co 7363 ∈ cmpo 7365 ωcom 7813 ↑_m cmap 8770 WUnicwun 10621 ndxcnx 17161 Basecbs 17177 SetCatcsetc 18040

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 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pr 5369

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 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-iun 4930 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7366 df-oprab 7367 df-mpo 7368

This theorem is referenced by: funcsetcestrclem9 18127 fthsetcestrc 18129 fullsetcestrc 18130

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