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Theorem fundcmpsurinjimaid 45693
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 6769 . . 3 (𝐹:𝐴𝐵𝐹:𝐴onto→(𝐹𝐴))
2 fundcmpsurinjimaid.g . . . . 5 𝐺 = (𝑥𝐴 ↦ (𝐹𝑥))
3 ffn 6672 . . . . . . 7 (𝐹:𝐴𝐵𝐹 Fn 𝐴)
4 dffn5 6905 . . . . . . 7 (𝐹 Fn 𝐴𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
53, 4sylib 217 . . . . . 6 (𝐹:𝐴𝐵𝐹 = (𝑥𝐴 ↦ (𝐹𝑥)))
65eqcomd 2739 . . . . 5 (𝐹:𝐴𝐵 → (𝑥𝐴 ↦ (𝐹𝑥)) = 𝐹)
72, 6eqtrid 2785 . . . 4 (𝐹:𝐴𝐵𝐺 = 𝐹)
8 eqidd 2734 . . . 4 (𝐹:𝐴𝐵𝐴 = 𝐴)
9 fundcmpsurinjimaid.i . . . . 5 𝐼 = (𝐹𝐴)
109a1i 11 . . . 4 (𝐹:𝐴𝐵𝐼 = (𝐹𝐴))
117, 8, 10foeq123d 6781 . . 3 (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐹:𝐴onto→(𝐹𝐴)))
121, 11mpbird 257 . 2 (𝐹:𝐴𝐵𝐺:𝐴onto𝐼)
13 f1oi 6826 . . 3 ( I ↾ 𝐼):𝐼1-1-onto𝐼
14 f1of1 6787 . . 3 (( I ↾ 𝐼):𝐼1-1-onto𝐼 → ( I ↾ 𝐼):𝐼1-1𝐼)
15 fundcmpsurinjimaid.h . . . . . . 7 𝐻 = ( I ↾ 𝐼)
16 f1eq1 6737 . . . . . . 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 6693 . . . . . 6 (𝐹:𝐴𝐵 → (𝐹𝐴) ⊆ 𝐵)
209, 19eqsstrid 3996 . . . . 5 (𝐹:𝐴𝐵𝐼𝐵)
21 f1ss 6748 . . . . 5 ((𝐻:𝐼1-1𝐼𝐼𝐵) → 𝐻:𝐼1-1𝐵)
2218, 20, 21syl2an 597 . . . 4 ((( I ↾ 𝐼):𝐼1-1𝐼𝐹:𝐴𝐵) → 𝐻:𝐼1-1𝐵)
2322ex 414 . . 3 (( I ↾ 𝐼):𝐼1-1𝐼 → (𝐹:𝐴𝐵𝐻:𝐼1-1𝐵))
2413, 14, 23mp2b 10 . 2 (𝐹:𝐴𝐵𝐻:𝐼1-1𝐵)
2515fveq1i 6847 . . . . 5 (𝐻‘(𝐹𝑥)) = (( I ↾ 𝐼)‘(𝐹𝑥))
263adantr 482 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝐹 Fn 𝐴)
27 simpr 486 . . . . . . . 8 ((𝐹:𝐴𝐵𝑥𝐴) → 𝑥𝐴)
2826, 27, 27fnfvimad 7188 . . . . . . 7 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ (𝐹𝐴))
2928, 9eleqtrrdi 2845 . . . . . 6 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐹𝑥) ∈ 𝐼)
30 fvresi 7123 . . . . . 6 ((𝐹𝑥) ∈ 𝐼 → (( I ↾ 𝐼)‘(𝐹𝑥)) = (𝐹𝑥))
3129, 30syl 17 . . . . 5 ((𝐹:𝐴𝐵𝑥𝐴) → (( I ↾ 𝐼)‘(𝐹𝑥)) = (𝐹𝑥))
3225, 31eqtrid 2785 . . . 4 ((𝐹:𝐴𝐵𝑥𝐴) → (𝐻‘(𝐹𝑥)) = (𝐹𝑥))
3332mpteq2dva 5209 . . 3 (𝐹:𝐴𝐵 → (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))) = (𝑥𝐴 ↦ (𝐹𝑥)))
342coeq2i 5820 . . . 4 (𝐻𝐺) = (𝐻 ∘ (𝑥𝐴 ↦ (𝐹𝑥)))
35 f1of 6788 . . . . . . . 8 (( I ↾ 𝐼):𝐼1-1-onto𝐼 → ( I ↾ 𝐼):𝐼𝐼)
3613, 35ax-mp 5 . . . . . . 7 ( I ↾ 𝐼):𝐼𝐼
3715feq1i 6663 . . . . . . 7 (𝐻:𝐼𝐼 ↔ ( I ↾ 𝐼):𝐼𝐼)
3836, 37mpbir 230 . . . . . 6 𝐻:𝐼𝐼
3938a1i 11 . . . . 5 (𝐹:𝐴𝐵𝐻:𝐼𝐼)
4039, 29cofmpt 7082 . . . 4 (𝐹:𝐴𝐵 → (𝐻 ∘ (𝑥𝐴 ↦ (𝐹𝑥))) = (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))))
4134, 40eqtrid 2785 . . 3 (𝐹:𝐴𝐵 → (𝐻𝐺) = (𝑥𝐴 ↦ (𝐻‘(𝐹𝑥))))
4233, 41, 53eqtr4rd 2784 . 2 (𝐹:𝐴𝐵𝐹 = (𝐻𝐺))
4312, 24, 423jca 1129 1 (𝐹:𝐴𝐵 → (𝐺:𝐴onto𝐼𝐻:𝐼1-1𝐵𝐹 = (𝐻𝐺)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wss 3914  cmpt 5192   I cid 5534  cres 5639  cima 5640  ccom 5641   Fn wfn 6495  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-sep 5260  ax-nul 5267  ax-pr 5388
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-ral 3062  df-rex 3071  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-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  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:  fundcmpsurinjALT  45694
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