Theorem swapf1vala 50343

Description: The object part of the swap functor. See also swapf1val 50344. (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 2762	. . . 4 ⊢ (𝜑 → (Hom ‘𝑆) = (Hom ‘𝑆))
6	1, 2, 3, 4, 5	swapfval 50339	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))⟩)
7	6	fveq2d 6887	. 2 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (1st ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))⟩))
8	4	fvexi 6897	. . . 4 ⊢ 𝐵 ∈ V
9	8	mptex 7227	. . 3 ⊢ (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}) ∈ V
10	8, 8	mpoex 8090	. . 3 ⊢ (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓})) ∈ V
11	9, 10	op1st 8007	. 2 ⊢ (1st ‘⟨(𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}), (𝑢 ∈ 𝐵, 𝑣 ∈ 𝐵 ↦ (𝑓 ∈ (𝑢(Hom ‘𝑆)𝑣) ↦ ∪ ◡{𝑓}))⟩) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥})
12	7, 11	eqtrdi 2812	1 ⊢ (𝜑 → (1st ‘(𝐶 swapF 𝐷)) = (𝑥 ∈ 𝐵 ↦ ∪ ◡{𝑥}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  {csn 4584  ⟨cop 4590  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5650  ‘cfv 6537  (class class class)co 7418   ∈ cmpo 7420  1^st c1st 7997  Basecbs 17380  Hom chom 17432   ×_c cxpc 18335   swap_F cswapf 50336

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7421  df-oprab 7422  df-mpo 7423  df-1st 7999  df-2nd 8000  df-swapf 50337

This theorem is used by:  swapf1val  50344