Theorem swapf1vala 49741

Description: The object part of the swap functor. See also swapf1val 49742. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapfval.c	⊢ (𝜑 → 𝐶 ∈ 𝑈)
swapfval.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
swapf2fvala.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf2fvala.b	⊢ 𝐵 = (Base‘𝑆)

Assertion

Ref	Expression
swapf1vala	⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝐷(𝑥)   𝑆(𝑥)   𝑈(𝑥)   𝑉(𝑥)

Proof of Theorem swapf1vala

Dummy variables 𝑢 𝑣 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		swapfval.c	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑈)
2		swapfval.d	. . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
3		swapf2fvala.s	. . . 4 ⊢ 𝑆 = (𝐶 ×c 𝐷)
4		swapf2fvala.b	. . . 4 ⊢ 𝐵 = (Base‘𝑆)
5		eqidd 2737	. . . 4 ⊢ (𝜑 → (Hom ‘𝑆) = (Hom ‘𝑆))
6	1, 2, 3, 4, 5	swapfval 49737	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))⟩)
7	6	fveq2d 6844	. 2 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (1st ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))⟩))
8	4	fvexi 6854	. . . 4 ⊢ 𝐵 ∈ V
9	8	mptex 7178	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}) ∈ V
10	8, 8	mpoex 8032	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) ∈ V
11	9, 10	op1st 7950	. 2 ⊢ (1st ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))⟩) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥})
12	7, 11	eqtrdi 2787	1 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4567  ⟨cop 4573  ∪ cuni 4850   ↦ cmpt 5166  ^◡ccnv 5630  ‘cfv 6498  (class class class)co 7367   ∈ cmpo 7369  1^st c1st 7940  Basecbs 17179  Hom chom 17231   ×_c cxpc 18134   swap_F cswapf 49734

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-swapf 49735

This theorem is referenced by:  swapf1val  49742