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Theorem fundcmpsurbijinj 45592
Description: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective, a bijective and an injective function. (Contributed by AV, 23-Mar-2024.)
Assertion
Ref Expression
fundcmpsurbijinj ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
Distinct variable groups:   𝐴,𝑔,,𝑝   𝐵,𝑔,,𝑝   𝑔,𝐹,,𝑝   𝑔,𝑉   𝐴,𝑖,𝑞,𝑔,,𝑝   𝐵,𝑖,𝑞   𝑖,𝐹,𝑞
Allowed substitution hints:   𝑉(,𝑖,𝑞,𝑝)

Proof of Theorem fundcmpsurbijinj
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffun 6671 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
2 funimaexg 6587 . . . 4 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
31, 2sylan 580 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (𝐹𝐴) ∈ V)
4 abrexexg 7893 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
54adantl 482 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
6 fveq2 6842 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
76sneqd 4598 . . . . . . . 8 (𝑦 = 𝑥 → {(𝐹𝑦)} = {(𝐹𝑥)})
87imaeq2d 6013 . . . . . . 7 (𝑦 = 𝑥 → (𝐹 “ {(𝐹𝑦)}) = (𝐹 “ {(𝐹𝑥)}))
98eqeq2d 2747 . . . . . 6 (𝑦 = 𝑥 → (𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑥)})))
109cbvrexvw 3226 . . . . 5 (∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}))
1110abbii 2806 . . . 4 {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1211fundcmpsurbijinjpreimafv 45589 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
13 foeq3 6754 . . . . . . . . 9 (𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
1413adantl 482 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
15 f1oeq23 6775 . . . . . . . . 9 ((𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ 𝑞 = (𝐹𝐴)) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
1615ancoms 459 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
17 f1eq2 6734 . . . . . . . . 9 (𝑞 = (𝐹𝐴) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1817adantr 481 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1914, 16, 183anbi123d 1436 . . . . . . 7 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → ((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ↔ (𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵)))
2019anbi1d 630 . . . . . 6 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))))
21203exbidv 1928 . . . . 5 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (∃𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))))
2221spc2egv 3558 . . . 4 (((𝐹𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V) → (∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))))
2322imp 407 . . 3 ((((𝐹𝐴) ∈ V ∧ {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V) ∧ ∃𝑔𝑖((𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ :{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴) ∧ 𝑖:(𝐹𝐴)–1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔))) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
243, 5, 12, 23syl21anc 836 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
25 exrot4 2166 . . 3 (∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
26 excom13 2164 . . . 4 (∃𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
27262exbii 1851 . . 3 (∃𝑔𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
2825, 27bitri 274 . 2 (∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
2924, 28sylib 217 1 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑔𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wrex 3073  Vcvv 3445  {csn 4586  ccnv 5632  cima 5636  ccom 5637  Fun wfun 6490  wf 6492  1-1wf1 6493  ontowfo 6494  1-1-ontowf1o 6495  cfv 6496
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504
This theorem is referenced by: (None)
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