Theorem swapf2f1oa 50730

Description: The morphism part of the swap functor is a bijection between hom-sets. (Contributed by Zhi Wang, 9-Oct-2025.)

Hypotheses

Ref	Expression
swapf1f1o.o	⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
swapf1f1o.s	⊢ 𝑆 = (𝐶 ×c 𝐷)
swapf1f1o.t	⊢ 𝑇 = (𝐷 ×c 𝐶)
swapf2f1o.h	⊢ 𝐻 = (Hom ‘𝑆)
swapf2f1o.j	⊢ 𝐽 = (Hom ‘𝑇)
swapf2f1oa.b	⊢ 𝐵 = (Base‘𝑆)
swapf2f1oa.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
swapf2f1oa.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
swapf2f1oa	⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)⟶1-1onto((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

Proof of Theorem swapf2f1oa

Step	Hyp	Ref	Expression
1		swapf1f1o.o	. . 3 ⊢ (𝜑 → (𝐶 swapF 𝐷) = ⟨𝑂, 𝑃⟩)
2		swapf1f1o.s	. . 3 ⊢ 𝑆 = (𝐶 ×c 𝐷)
3		swapf1f1o.t	. . 3 ⊢ 𝑇 = (𝐷 ×c 𝐶)
4		swapf2f1o.h	. . 3 ⊢ 𝐻 = (Hom ‘𝑆)
5		swapf2f1o.j	. . 3 ⊢ 𝐽 = (Hom ‘𝑇)
6		swapf2f1oa.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7		swapf2f1oa.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝑆)
8		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		eqid 2763	. . . . . . 7 ⊢ (Base‘𝐷) = (Base‘𝐷)
10	2, 8, 9	xpcbas 18402	. . . . . 6 ⊢ ((Base‘𝐶) × (Base‘𝐷)) = (Base‘𝑆)
11	7, 10	eqtr4i 2789	. . . . 5 ⊢ 𝐵 = ((Base‘𝐶) × (Base‘𝐷))
12	6, 11	eleqtrdi 2874	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)))
13		xp1st 8067	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑋) ∈ (Base‘𝐶))
14	12, 13	syl 18	. . 3 ⊢ (𝜑 → (1st ‘𝑋) ∈ (Base‘𝐶))
15		xp2nd 8068	. . . 4 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd ‘𝑋) ∈ (Base‘𝐷))
16	12, 15	syl 18	. . 3 ⊢ (𝜑 → (2nd ‘𝑋) ∈ (Base‘𝐷))
17		swapf2f1oa.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
18	17, 11	eleqtrdi 2874	. . . 4 ⊢ (𝜑 → 𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)))
19		xp1st 8067	. . . 4 ⊢ (𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (1st ‘𝑌) ∈ (Base‘𝐶))
20	18, 19	syl 18	. . 3 ⊢ (𝜑 → (1st ‘𝑌) ∈ (Base‘𝐶))
21		xp2nd 8068	. . . 4 ⊢ (𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)) → (2nd ‘𝑌) ∈ (Base‘𝐷))
22	18, 21	syl 18	. . 3 ⊢ (𝜑 → (2nd ‘𝑌) ∈ (Base‘𝐷))
23	1, 2, 3, 4, 5, 14, 16, 20, 22	swapf2f1o 50729	. 2 ⊢ (𝜑 → (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑌), (2nd ‘𝑌)⟩):(⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝐻⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)⟶1-1onto(⟨(2nd ‘𝑋), (1st ‘𝑋)⟩𝐽⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
24		1st2nd2 8074	. . . . 5 ⊢ (𝑋 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
25	12, 24	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 = ⟨(1st ‘𝑋), (2nd ‘𝑋)⟩)
26		1st2nd2 8074	. . . . 5 ⊢ (𝑌 ∈ ((Base‘𝐶) × (Base‘𝐷)) → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
27	18, 26	syl 18	. . . 4 ⊢ (𝜑 → 𝑌 = ⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)
28	25, 27	oveq12d 7469	. . 3 ⊢ (𝜑 → (𝑋𝑃𝑌) = (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑌), (2nd ‘𝑌)⟩))
29	25, 27	oveq12d 7469	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑌) = (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝐻⟨(1st ‘𝑌), (2nd ‘𝑌)⟩))
30	1, 2, 7, 6	swapf1a 50722	. . . 4 ⊢ (𝜑 → (𝑂‘𝑋) = ⟨(2nd ‘𝑋), (1st ‘𝑋)⟩)
31	1, 2, 7, 17	swapf1a 50722	. . . 4 ⊢ (𝜑 → (𝑂‘𝑌) = ⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)
32	30, 31	oveq12d 7469	. . 3 ⊢ (𝜑 → ((𝑂‘𝑋)𝐽(𝑂‘𝑌)) = (⟨(2nd ‘𝑋), (1st ‘𝑋)⟩𝐽⟨(2nd ‘𝑌), (1st ‘𝑌)⟩))
33	28, 29, 32	f1oeq123d 6848	. 2 ⊢ (𝜑 → ((𝑋𝑃𝑌):(𝑋𝐻𝑌)⟶1-1onto((𝑂‘𝑋)𝐽(𝑂‘𝑌)) ↔ (⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝑃⟨(1st ‘𝑌), (2nd ‘𝑌)⟩):(⟨(1st ‘𝑋), (2nd ‘𝑋)⟩𝐻⟨(1st ‘𝑌), (2nd ‘𝑌)⟩)⟶1-1onto(⟨(2nd ‘𝑋), (1st ‘𝑋)⟩𝐽⟨(2nd ‘𝑌), (1st ‘𝑌)⟩)))
34	23, 33	mpbird 260	1 ⊢ (𝜑 → (𝑋𝑃𝑌):(𝑋𝐻𝑌)⟶1-1onto((𝑂‘𝑋)𝐽(𝑂‘𝑌)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ⟨cop 4840   × cxp 5656  ⟶1-1ontowf1o 6565  ‘cfv 6566  (class class class)co 7451  1^st c1st 8033  2^nd c2nd 8034  Basecbs 17431  Hom chom 17483   ×_c cxpc 18392   swap_F cswapf 50712

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783  ax-cnex 11295  ax-resscn 11296  ax-1cn 11297  ax-icn 11298  ax-addcl 11299  ax-addrcl 11300  ax-mulcl 11301  ax-mulrcl 11302  ax-mulcom 11303  ax-addass 11304  ax-mulass 11305  ax-distr 11306  ax-i2m1 11307  ax-1ne0 11308  ax-1rid 11309  ax-rnegex 11310  ax-rrecex 11311  ax-cnre 11312  ax-pre-lttri 11313  ax-pre-lttrn 11314  ax-pre-ltadd 11315  ax-pre-mulgt0 11316

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-pred 6330  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-frecs 8337  df-wrecs 8368  df-recs 8417  df-rdg 8456  df-1o 8514  df-er 8755  df-en 9012  df-dom 9013  df-sdom 9014  df-fin 9015  df-pnf 11384  df-mnf 11385  df-xr 11386  df-ltxr 11387  df-le 11388  df-sub 11582  df-neg 11583  df-nn 12376  df-2 12445  df-3 12446  df-4 12447  df-5 12448  df-6 12449  df-7 12450  df-8 12451  df-9 12452  df-n0 12647  df-z 12734  df-dec 12855  df-uz 13006  df-fz 13680  df-struct 17369  df-slot 17404  df-ndx 17416  df-base 17432  df-hom 17497  df-cco 17498  df-xpc 18396  df-swapf 50713

This theorem is used by:  swapfcoa  50734  swapffunc  50735  swapfffth  50736