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Theorem fundcmpsurinjimaid 44863
Description: Every function 𝐹:𝐴𝐵 can be decomposed into a surjective function onto the image (𝐹𝐴) of the domain of 𝐹 and an injective function from the image (𝐹𝐴). (Contributed by AV, 17-Mar-2024.)
Hypotheses
Ref Expression
fundcmpsurinjimaid.i 𝐼 = (𝐹𝐴)
fundcmpsurinjimaid.g 𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))
fundcmpsurinjimaid.h 𝐻 = ( I ↾ 𝐼)
Assertion
Ref Expression
fundcmpsurinjimaid (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹   𝑥,𝐻   𝑥,𝐼
Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem fundcmpsurinjimaid
StepHypRef Expression
1 fimadmfo 6697 . . 3 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
2 fundcmpsurinjimaid.g . . . . 5 𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))
3 ffn 6600 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
4 dffn5 6828 . . . . . . 7 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
53, 4sylib 217 . . . . . 6 (𝐹:𝐴𝐵𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
65eqcomd 2744 . . . . 5 (𝐹:𝐴𝐵 → (𝑥𝐴 ↦ (𝐹𝑥)) = 𝐹)
72, 6eqtrid 2790 . . . 4 (𝐹:𝐴𝐵𝐺 = 𝐹)
8 eqidd 2739 . . . 4 (𝐹:𝐴𝐵𝐴 = 𝐴)
9 fundcmpsurinjimaid.i . . . . 5 𝐼 = (𝐹𝐴)
109a1i 11 . . . 4 (𝐹:𝐴𝐵𝐼 = (𝐹𝐴))
117, 8, 10foeq123d 6709 . . 3 (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐹:𝐴onto→(𝐹𝐴)))
121, 11mpbird 256 . 2 (𝐹:𝐴𝐵𝐺:𝐴onto𝐼)
13 f1oi 6754 . . 3 ( I ↾ 𝐼):𝐼1-1-onto𝐼
14 f1of1 6715 . . 3 (( I ↾ 𝐼):𝐼1-1-onto𝐼 → ( I ↾ 𝐼):𝐼1-1𝐼)
15 fundcmpsurinjimaid.h . . . . . . 7 𝐻 = ( I ↾ 𝐼)
16 f1eq1 6665 . . . . . . 7 (𝐻 = ( I ↾ 𝐼) → (𝐻:𝐼1-1𝐼 ↔ ( I ↾ 𝐼):𝐼1-1𝐼))
1715, 16ax-mp 5 . . . . . 6 (𝐻:𝐼1-1𝐼 ↔ ( I ↾ 𝐼):𝐼1-1𝐼)
1817biimpri 227 . . . . 5 (( I ↾ 𝐼):𝐼1-1𝐼𝐻:𝐼1-1𝐼)
19 fimass 6621 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐴) ⊆ 𝐵)
209, 19eqsstrid 3969 . . . . 5 (𝐹:𝐴𝐵𝐼𝐵)
21 f1ss 6676 . . . . 5 ((𝐻:𝐼1-1𝐼𝐼𝐵) → 𝐻:𝐼1-1𝐵)
2218, 20, 21syl2an 596 . . . 4 ((( I ↾ 𝐼):𝐼1-1𝐼𝐹:𝐴𝐵) → 𝐻:𝐼1-1𝐵)
2322ex 413 . . 3 (( I ↾ 𝐼):𝐼1-1𝐼 → (𝐹:𝐴𝐵𝐻:𝐼1-1𝐵))
2413, 14, 23mp2b 10 . 2 (𝐹:𝐴𝐵𝐻:𝐼1-1𝐵)
2515fveq1i 6775 . . . . 5 (𝐻‘(𝐹𝑥)) = (( I ↾ 𝐼)‘(𝐹𝑥))
263adantr 481 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝐹 Fn 𝐴)
27 simpr 485 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
2826, 27, 27fnfvimad 7110 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2928, 9eleqtrrdi 2850 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐼)
30 fvresi 7045 . . . . . 6 ((𝐹𝑥) ∈ 𝐼 → (( I ↾ 𝐼)‘(𝐹𝑥)) = (𝐹𝑥))
3129, 30syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐼)‘(𝐹𝑥)) = (𝐹𝑥))
3225, 31eqtrid 2790 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐻‘(𝐹𝑥)) = (𝐹𝑥))
3332mpteq2dva 5174 . . 3 (𝐹:𝐴𝐵 → (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))) = (𝑥𝐴 ↦ (𝐹𝑥)))
342coeq2i 5769 . . . 4 (𝐻𝐺) = (𝐻 ∘ (𝑥𝐴 ↦ (𝐹𝑥)))
35 f1of 6716 . . . . . . . 8 (( I ↾ 𝐼):𝐼1-1-onto𝐼 → ( I ↾ 𝐼):𝐼𝐼)
3613, 35ax-mp 5 . . . . . . 7 ( I ↾ 𝐼):𝐼𝐼
3715feq1i 6591 . . . . . . 7 (𝐻:𝐼𝐼 ↔ ( I ↾ 𝐼):𝐼𝐼)
3836, 37mpbir 230 . . . . . 6 𝐻:𝐼𝐼
3938a1i 11 . . . . 5 (𝐹:𝐴𝐵𝐻:𝐼𝐼)
4039, 29cofmpt 7004 . . . 4 (𝐹:𝐴𝐵 → (𝐻 ∘ (𝑥𝐴 ↦ (𝐹𝑥))) = (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))))
4134, 40eqtrid 2790 . . 3 (𝐹:𝐴𝐵 → (𝐻𝐺) = (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))))
4233, 41, 53eqtr4rd 2789 . 2 (𝐹:𝐴𝐵𝐹 = (𝐻𝐺))
4312, 24, 423jca 1127 1 (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2106  wss 3887  cmpt 5157   I cid 5488  cres 5591  cima 5592  ccom 5593   Fn wfn 6428  wf 6429  1-1wf1 6430  ontowfo 6431  1-1-ontowf1o 6432  cfv 6433
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441
This theorem is referenced by:  fundcmpsurinjALT  44864
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