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Theorem fundcmpsurbijinj 45692
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 6675 . . . 4 (𝐹:𝐴𝐵 → Fun 𝐹)
2 funimaexg 6591 . . . 4 ((Fun 𝐹𝐴𝑉) → (𝐹𝐴) ∈ V)
31, 2sylan 581 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → (𝐹𝐴) ∈ V)
4 abrexexg 7897 . . . 4 (𝐴𝑉 → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
54adantl 483 . . 3 ((𝐹:𝐴𝐵𝐴𝑉) → {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∈ V)
6 fveq2 6846 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
76sneqd 4602 . . . . . . . 8 (𝑦 = 𝑥 → {(𝐹𝑦)} = {(𝐹𝑥)})
87imaeq2d 6017 . . . . . . 7 (𝑦 = 𝑥 → (𝐹 “ {(𝐹𝑦)}) = (𝐹 “ {(𝐹𝑥)}))
98eqeq2d 2744 . . . . . 6 (𝑦 = 𝑥 → (𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ 𝑧 = (𝐹 “ {(𝐹𝑥)})))
109cbvrexvw 3225 . . . . 5 (∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)}) ↔ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)}))
1110abbii 2803 . . . 4 {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} = {𝑧 ∣ ∃𝑥𝐴 𝑧 = (𝐹 “ {(𝐹𝑥)})}
1211fundcmpsurbijinjpreimafv 45689 . . 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 6758 . . . . . . . . 9 (𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
1413adantl 483 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑔:𝐴onto𝑝𝑔:𝐴onto→{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}))
15 f1oeq23 6779 . . . . . . . . 9 ((𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})} ∧ 𝑞 = (𝐹𝐴)) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
1615ancoms 460 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (:𝑝1-1-onto𝑞:{𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}–1-1-onto→(𝐹𝐴)))
17 f1eq2 6738 . . . . . . . . 9 (𝑞 = (𝐹𝐴) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1817adantr 482 . . . . . . . 8 ((𝑞 = (𝐹𝐴) ∧ 𝑝 = {𝑧 ∣ ∃𝑦𝐴 𝑧 = (𝐹 “ {(𝐹𝑦)})}) → (𝑖:𝑞1-1𝐵𝑖:(𝐹𝐴)–1-1𝐵))
1914, 16, 183anbi123d 1437 . . . . . . 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 631 . . . . . 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 1929 . . . . 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 3560 . . . 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 408 . . 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 837 . 2 ((𝐹:𝐴𝐵𝐴𝑉) → ∃𝑞𝑝𝑔𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
25 exrot4 2167 . . 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 2165 . . . 4 (∃𝑞𝑝𝑖((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)) ↔ ∃𝑖𝑝𝑞((𝑔:𝐴onto𝑝:𝑝1-1-onto𝑞𝑖:𝑞1-1𝐵) ∧ 𝐹 = ((𝑖) ∘ 𝑔)))
27262exbii 1852 . . 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 275 . 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 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3070  Vcvv 3447  {csn 4590  ccnv 5636  cima 5640  ccom 5641  Fun wfun 6494  wf 6496  1-1wf1 6497  ontowfo 6498  1-1-ontowf1o 6499  cfv 6500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508
This theorem is referenced by: (None)
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