Theorem 3f1oss2 46981

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 6869	. . 3 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → ◡𝐹:𝐵–1-1-onto→𝐴)
2		id 22	. . 3 ⊢ (𝐺:𝐶–1-1-onto→𝐷 → 𝐺:𝐶–1-1-onto→𝐷)
3		f1ocnv 6869	. . 3 ⊢ (𝐻:𝐸–1-1-onto→𝐼 → ◡𝐻:𝐼–1-1-onto→𝐸)
4		3f1oss1 46980	. . 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 1415	. 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 6860	. . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → Rel 𝐹)
7		dfrel2 6215	. . . . . . . . 9 ⊢ (Rel 𝐹 ↔ ◡◡𝐹 = 𝐹)
8	7	biimpi 216	. . . . . . . 8 ⊢ (Rel 𝐹 → ◡◡𝐹 = 𝐹)
9	8	eqcomd 2746	. . . . . . 7 ⊢ (Rel 𝐹 → 𝐹 = ◡◡𝐹)
10	9	coeq2d 5882	. . . . . 6 ⊢ (Rel 𝐹 → ((◡𝐻 ∘ 𝐺) ∘ 𝐹) = ((◡𝐻 ∘ 𝐺) ∘ ◡◡𝐹))
11	10	f1oeq1d 6852	. . . . 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 17	. . . 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 1133	. . 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 480	. 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 257	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))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1537   ⊆ wss 3976  ^◡ccnv 5694   “ cima 5698   ∘ ccom 5699  Rel wrel 5700  –1-1-onto→wf1o 6567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-iota 6520  df-fun 6570  df-fn 6571  df-f 6572  df-f1 6573  df-fo 6574  df-f1o 6575  df-fv 6576

This theorem is referenced by:  uspgrlim  47806