Theorem 3f1oss2 48253

Description: The composition of three bijections as bijection from the image of the converse of the domain onto the image of the converse of the range of the middle bijection. (Contributed by AV, 15-Aug-2025.)

Assertion

Ref	Expression
3f1oss2	⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))

Proof of Theorem 3f1oss2

Step	Hyp	Ref	Expression
1		f1ocnv 6840	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		id 23	. . 3 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1-onto→𝐷)
3		f1ocnv 6840	. . 3 ⊢ (𝐻:𝐸–1-1-onto→𝐼 → ◡𝐻:𝐼–1-1-onto→𝐸)
4		3f1oss1 48252	. . 3 ⊢ (((◡𝐹:𝐵–1-1-onto→𝐴 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ ◡𝐻:𝐼–1-1-onto→𝐸) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))
5	1, 2, 3, 4	syl3anl 1442	. 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))
6		f1orel 6830	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
7		dfrel2 6182	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	7	biimpi 219	. . . . . . . 8 ⊢ (Rel 𝐹 → ◡◡𝐹 = 𝐹)
9	8	eqcomd 2766	. . . . . . 7 ⊢ (Rel 𝐹 → 𝐹 = ◡◡𝐹)
10	9	coeq2d 5840	. . . . . 6 ⊢ (Rel 𝐹 → ((◡𝐻 ∘ 𝐺) ∘ 𝐹) = ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹))
11	10	f1oeq1d 6822	. . . . 5 ⊢ (Rel 𝐹 → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)))
12	6, 11	syl 18	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)))
13	12	3ad2ant1 1151	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)))
14	13	adantr 486	. 2 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → (((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷) ↔ ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷)))
15	5, 14	mpbird 260	1 ⊢ (((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝐺:𝐶–1-1-onto→𝐷 ∧ 𝐻:𝐸–1-1-onto→𝐼) ∧ (𝐶 ⊆ 𝐵 ∧ 𝐷 ⊆ 𝐼)) → ((◡𝐻 ∘ 𝐺) ∘ 𝐹):(◡𝐹 “ 𝐶)–1-1-onto→(◡𝐻 “ 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ⊆ wss 3896  ^◡ccnv 5649   “ cima 5653   ∘ ccom 5654  Rel wrel 5655  –1-1-onto→wf1o 6540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3848  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6496  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549

This theorem is used by:  uspgrlim  49197