Theorem swapf2a 49630

Description: The morphism part of the swap functor swaps the morphisms. (Contributed by Zhi Wang, 7-Oct-2025.)

Hypotheses

Ref	Expression
swapf1a.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
swapf1a.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf1a.b	⊢ 𝐵 = (Base‘𝑆)
swapf1a.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapf2a.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
swapf2a.h	⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
swapf2a.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))

Assertion

Ref	Expression
swapf2a	⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

Proof of Theorem swapf2a

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		swapf1a.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
2		swapf1a.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
3		swapf1a.b	. . 3 ⊢ 𝐵 = (Base‘𝑆)
4		swapf1a.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		swapf2a.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		swapf2a.h	. . 3 ⊢ (𝜑 → 𝐻 = (Hom ‘𝑆))
7	1, 2, 3, 4, 5, 6	swapf2vala 49629	. 2 ⊢ (𝜑 → (𝑋𝑃𝑌) = (𝑓 ∈ (𝑋𝐻𝑌) ↦ ∪ ◡{𝑓}))
8		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
9	8	sneqd 4594	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → {𝑓} = {𝐹})
10	9	cnveqd 5832	. . . 4 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ◡{𝑓} = ◡{𝐹})
11	10	unieqd 4878	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ∪ ◡{𝐹})
12		swapf2a.f	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
13	6	oveqd 7385	. . . . . . 7 ⊢ (𝜑 → (𝑋𝐻𝑌) = (𝑋(Hom ‘𝑆)𝑌))
14		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
15		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
16		eqid 2737	. . . . . . . 8 ⊢ (Hom ‘𝑆) = (Hom ‘𝑆)
17	2, 3, 14, 15, 16, 4, 5	xpchom 18115	. . . . . . 7 ⊢ (𝜑 → (𝑋(Hom ‘𝑆)𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
18	13, 17	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → (𝑋𝐻𝑌) = (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
19	12, 18	eleqtrd 2839	. . . . 5 ⊢ (𝜑 → 𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))))
20		2nd1st 7992	. . . . 5 ⊢ (𝐹 ∈ (((1st ‘𝑋)(Hom ‘𝐶)(1st ‘𝑌)) × ((2nd ‘𝑋)(Hom ‘𝐷)(2nd ‘𝑌))) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
21	19, 20	syl 17	. . . 4 ⊢ (𝜑 → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
22	21	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝐹} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
23	11, 22	eqtrd 2772	. 2 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → ∪ ◡{𝑓} = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)
24		opex 5419	. . 3 ⊢ ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V
25	24	a1i 11	. 2 ⊢ (𝜑 → ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩ ∈ V)
26	7, 23, 12, 25	fvmptd 6957	1 ⊢ (𝜑 → ((𝑋𝑃𝑌)‘𝐹) = ⟨(2nd ‘𝐹), (1st ‘𝐹)⟩)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582  ⟨cop 4588  ∪ cuni 4865   × cxp 5630  ^◡ccnv 5631  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148  Hom chom 17200   ×_c cxpc 18103   swap_F cswapf 49618

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 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-tp 4587  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-om 7819  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-1o 8407  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-fin 8899  df-pnf 11180  df-mnf 11181  df-xr 11182  df-ltxr 11183  df-le 11184  df-sub 11378  df-neg 11379  df-nn 12158  df-2 12220  df-3 12221  df-4 12222  df-5 12223  df-6 12224  df-7 12225  df-8 12226  df-9 12227  df-n0 12414  df-z 12501  df-dec 12620  df-uz 12764  df-fz 13436  df-struct 17086  df-slot 17121  df-ndx 17133  df-base 17149  df-hom 17213  df-cco 17214  df-xpc 18107  df-swapf 49619

This theorem is referenced by:  swapfcoa  49640